Binomial expansion triangle. +. The nth row in the triangle gives the This formula is known as the binomial theorem. Expand each binomial expression. To skip ahead: 1) for HOW TO EXPAND a BINOMIAL raised to a Example 6: Using Pascal’s Triangle to Find Binomial Expansions. We can use this, along with what we know about binomial coefficients, to give the general binomial expansion formula. The powers of the variable in the second term ascend in an orderly fashion. We can easily find the expansion of (x + y)2, (x + y)3, and others but finding the expansion of (x + y)21 is a tedious task and this task can easily be achieved using the Binomial Theorem or Binomial Expansion. Generate the seventh, eighth, and ninth rows of Pascal’s triangle. mc-TY-pascal-2009-1. Specifically, the binomial coefficient, typically written as , tells us the b th entry of the n th row of Pascal's triangle; n in Pascal's triangle indicates the row of the triangle starting at 0 from the top row; b indicates a coefficient The triangle can grow for as many rows as you desire, but the work becomes more tedious as the rows increase. Once again, we can see this as a block of 4. Notice the pattern in the triangle. To find any binomial coefficient, we need the two coefficients just above it. Get the free "Binomial Expansion Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. The exponent of a a decreases by 1 1, from n n to 0 0. com/JasonGibsonMathIn this lesson, you will learn about Pascal's Triangle, which is a pat Oct 6, 2021 · Figure 9. We will begin by finding the binomial coefficient. This very well-known connection is pointed out by the identity , where the binomial coefficients can be obtained by using Pascal's triangle. This is the bottom-most row, with coefficients 1 − 10 −45 − −45 −10 − 1. (x + y) 0. The coefficients are symmetric. Pascal’s Triangle. In mathematics, Pascal's triangle is a triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. com Pascal’s triangle (1653) has been found in the works of mathematicians dating back before the 2nd century BC. youtube. (x + y) 1. It shows all the expansions from `n=0` up to the power you have chosen. Pascal's triangle. The powers on x x begin with n n and decrease to 0. org right now: https://www. We will use the simple binomial a+b, but it could be any binomial. 二项式定理 （英語： Binomial theorem ）描述了 二项式 的 幂 的 代数 展开。. To generate Pascal’s Triangle, we start by writing a 1. So, our binomial expansion will have 10 +1 = 11 terms. The binomial theorem is the method of expanding an expression that has been raised to any finite power. Using Pascal's Triangle, you can quickly determine the coefficients of the binomial expansion without computing the binomial coefficients directly. (Free online tool expands any binomial expression) Precalculus The Binomial Theorem Pascal's Triangle and Binomial Expansion. You should know that (a + b)² = a² + 2ab + b² and you should be able to work out that (a + b)³ = a³ + 3a²b + 3b²a + b³ . In algebra and other branches of mathematics, Pascal’s triangle is a triangular array of numbers that lists the coefficients of the expansion of any binomial expression (x + y) n , where The binomial theorem (or binomial expansion) is a result of expanding the powers of binomials or sums of two terms. Find the coefficient of in the expansion of . com/playlist?list=PL5pdglZEO3NjsFjBEf0mu1u9Q1- The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. While Pascal’s triangle is useful in many different mathematical settings, it will be applied to the expansion of binomials. 除边缘的数字外，其他每一个数都为其上方两数之和。. The demonstration below illustrates the pattern. Since n = 13 and k = 10, Sep 26, 2016 · Explanation: (x +3y)4. Using Pascal’s triangle to expand a binomial expression. The coefficients of the terms in the expansion are the binomial coefficients \binom {n} {k} (kn). Solution: Using the triangle the coefficients for this expansion are 1, 4, 6, 4, and 1. david36. A binomial expression is the sum or difference of two terms. In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. Binomial Expression: A binomial expression is an algebraic expression that contains two dissimilar terms. Binomial Expansion Calculator. Flag. Sep 18, 2016 · 1 Answer. 1 color (blue) (1. In Row 6, for example, 15 is the sum of 5 and 10, and 20 is the sum of Video transcript. 2. com/watch?v=_3eqiXRwKXsLearn how to expand a binomial expression using Pascal's triangle. (x + 3) +. 1. It tells you the coefficients of the progressive terms In this paper we have effectively use the Pascal's Triangle and Binomial theorem to find ordinary decimal expansion of the number of the form n^k, where k is any positive integer. Pascal’s triangle is an arrangement of the binomial coefficients and one of the most known integer models. There are n + 1 n + 1 terms in the expansion of ( x + y) n. Key Questions. en. Mar 16, 2017 · More resources available at www. Let us start with an exponent of 0 and build upwards. 10 years ago. The powers variable in the first term of the binomial descend in an orderly fashion. Use this in conjunction with the binomial theorem to streamline the process of expanding binomials raised to powers. So the coefficient of is. Example 1. back to top . Each term has a combined degree of 5. Properties of Binomial Expansion. If n is very large, then it is very difficult to find the coefficients. Although Pascal discovered it independently, it had been observed in many cultures (from all around the world) before him. We now search for the row in the triangle with 11 terms. When a binomial expression is ra Jul 5, 2023 · Expanding binomials using Pascal's Triangle or Binomials Theorem can be very helpful when expanding it to a high power. Figure %: Pascal's Triangle. , a + b, a 3 + b 3, etc. Once the pattern is learned and unders Pascal's Triangle for a binomial expansion calculator negative power. For example, the expansion of \( (a+b)^4 \) can be read from the fifth row of Pascal's Triangle: $$ a^4 + 4a^3 b+6a^2 b^2 + 4ab^3 + b^4 $$ The Binomial Theorem can also be extended to negative and Pascal’s triangle and the binomial theorem. The coefficient function was a really tough one. If we want to raise a binomial expression to a power higher than 2 it is very cumbersome to do this by repeatedly multiplying x+1 or 3x+2y by itself. Find more Mathematics widgets in Wolfram|Alpha. Send feedback | Visit Wolfram|Alpha. Download Free PDF. The coefficients of the binomial expansion can be found from the pascals triangle or using the combinations formula of n C r = n! / [r! (n - r)!]. The formula for Pascal's Triangle comes from a relationship that you yourself might be able to see in the coefficients below. According to the theorem, we have. Learn how to use the binomial expansions theorem to expand a binomial and find any term or coefficient in this free math video by Mario's Math Tutoring. 二项式系数 出现在 杨辉三角 （帕斯卡三角）中。. So, the given numbers are the outcome of calculating the coefficient formula for each term. Here are the first five binomial expansions with their Notice that when we expanded (p + q) 4 (p + q) 4 in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal’s Triangle. (x + y) 4. In addition to the observation that the coefficients of a binomial expansion are the entries that create Pascal's triangle, there are several other interesting patterns and observations regarding the expansion of (a + b) n. Question 2. And I'm going to do multiple colors. Eg. These numbers are the coefficients of the terms in the binomial expansion. In the row below, row 2, we write two 1 ′ s. Related Symbolab blog posts 2 days ago · Pascal’s Triangle is a numerical pattern arranged in a triangular form. 根据该定理，可以将两个数之和的整数次幂诸如 展开为类似 项之和的 恒等式 Mar 7, 2011 · Fullscreen. For example, x+1 and 3x+2y are both binomial expressions. The coefficient of the term is 2560. So the row here is the line of the number 1’s on the In algebra, the binomial expansion and Pascal’s triangle are considered important. x + y. 1. Let us understand this with an example. (5 x-2 y)^ {4} This page titled 2. Binomial Expansion. Therefore, the number of terms is 9 + 1 = 10. For example, x+1, 3x+2y, a− b are all binomial expressions. Here you will explore patterns with binomial and polynomial expansion and find out how to get coefficients using Pascal’s Triangle. Practice. 4) Coefficient of b in expansion of (3 + b)4 108 5) Coefficient of x3y2 in expansion of (x − 3y)5 90 6) Coefficient of a2 in expansion of (2a + 1)5 40 Find each term described. To solve the above problems we can use combinations and factorial notation to help us expand binomial expressions. The power of the binomial is 9. This is Pascal’s triangle29; it provides a quick method for calculating the binomial coefficients. Exponent of 2 Feb 13, 2022 · 15. For larger indices, it is quicker than using the Pascal’s Triangle. An exercise in chapter 2 of Spivak's Calculus (4th ed. for all natural numbers n. If we want to raise a binomial expression to a power higher than 2 (for example if we want to find (x+1)7) it is very cumbersome to do this by Binomial Expansion. When the binomial is a difference, we must be careful in identifying the values we will use in the pattern. But why is that? Why are the coefficients related to combinations? Pascal's Triangle is probably the easiest way to expand binomials. Though it was named after the French scientist Blaise Pascal, it was studied in ancient India [1, 2], Persia [3, 4], China [5], Germany, and Italy [6]. The next example, the binomial is a difference. Expand . com/watch?v=qVsYE_oq-zQYOUTUBE CHANNEL at Sep 5, 2016 · Here is Binomial Expansion with a negative termhttps://www. And here comes Pascal's triangle. Note the exponents on the x start at 4 and decrease and the exponent on 3y starts at 0 and increases. misterwootube. It should also be obvious to you that (a + b)¹ = a + b . the second row in Pascal’s triangle represents the coefficients in (x+y) 2 and so on. View PDF. The signs for each term are going to alternate, because of the negative sign. 4. Contributed by: Pablo Alberca Bjerregaard (University of Malaga, Spain) (March 2011) In this video you will learn Pascal's Triangle. Binomial Theorem is composed of 2 function, one function gives you the coefficient of the member (the number of ways to get that member) and the other gives you the member. The Binomial Theorem builds on Pascal's Triangle in practical terms, since writing out triangles of numbers has its limits. We know the expansion of (x+y) 2 is x 2 + 2xy + y 2. ) talks about how Pascal's triangle gives the binomial coefficients. Work the problems out on paper. For nonzero real numbers a and b, (a + b)n = n ∑ j = 0(n j)an − jbj. https://www. e. One very clever and easy way to compute the coefficients of a binomial expansion is to use a triangle that starts with "1" at the top, then "1" and "1" at the second row. Consider Pascal's Triangle, as shown in the following diagram: The expansion of the above binomial will have n + 1 terms, in (A + B)n. Let us learn more about the binomial expansion formula. The Binomial Theorem The Binomial Theorem is a formula that can be used to expand any binomial. For example if you had (x + y) 4 the coefficients of each of the xy terms are the same as the numbers in row 4 of the triangle: 1, 4, 6, 4, 1. Pascal and combinations. What Is the Constant Term in the Binomial Theorem? The constant term in the binomial expansion is a numeric value and is independent of the variables. The top row is row zero, the next row is row 1, etc. This Demonstration illustrates the direct relation between Pascal's triangle and the binomial theorem. The triangle can be used to calculate the coefficients of the expansion of (a+b)n ( a + b) n by taking the exponent n n and adding 1 1. In Pascals Triangle, each entry is the sum of the two entries above it. The rows of Pascal's triangle contain the coefficients of binomial expansions and provide an alternate way to expand binomials. To get a feel of what this theorem is saying and how it really isn’t as hard to remember as it may first appear, let’s consider the specific case of n = 4. The row of Pascal's triangle starting 1, 6 gives the sequence of coefficients for the binomial expansion. 7) 2nd term in expansion of (y − 2x)4 −8y3x 8) 4th term in expansion of (4y + x)4 16 yx3 9) 1st term in expansion of (a + b)5 a5 10) 2nd term in expansion of (y Coefficients in the Binomial expansion can be found using Pascal's triangle as shown here. Sometimes we are interested only in a certain term of a binomial expansion. Find the tenth term of the expansion ( x + y) 13. These numbers will be the exponents of the variables, and you will consider the sum of a^ib^j with some coefficients. Exponent of 0. Therefore the term required is. + n C n x 0 y n. i. Row 4 is 1,4,6,4,1. If you like our videos follow us on Ins About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright Dec 22, 2017 · Triangle we want. Exercise 1. A binomial theorem is a powerful tool of expansion which has applications in Algebra, probability, etc. Pascals triangle can also be used to find the coefficient of the terms in the binomial expansion. Somewhere in our algebra studies, we learn that coefficients in a binomial expansion are rows from Pascal's triangle, or, equivalently, (x + y) n = n C 0 x n y 0 + n C 1 x n - 1 y 1 + …. This triangle provides the coefficients for the expansion of any binomial expression, with numbers organized in a way that they form a triangular shape. But why is that? Why are the coefficients related to combinations? The binomial has two properties that can help us to determine the coefficients of the remaining terms. The expansion follows the rule (a+b)n Pascal's Triangle is a triangle in which each row has one more entry than the preceding row, each row begins and ends with "1," and the interior elements are found by adding the adjacent elements in the preceding row. We w 1. Use row 4 of Pascal's triangle, shown above. Each number is the sum of the two numbers above it. Pascal's Triangle presents a formula that allows you to create the coefficients of the terms in a binomial expansion. Fully expand the expression (2 + 3 𝑥) . It's much simpler to use than the Binomial Theorem, which provides a formula for expanding binomials. x 4 + 4x 3 Y + 6x 2 Y 2 + 4XY 3 + Y 4. When an exponent is 0, we get 1: (a+b) 0 = 1. comTwitter: https://twitter. So far we have only seen how to expand (1+x)^{n}, but ideally we want a way to expand more general things, of the form (a+b)^{n}. Exponent of 1. The coefficients for varying x and y can be arranged to form Pascal's triangle. The larger the power is, the harder it is to expand expressions like this directly. When the exponent is 1, we get the original value, unchanged: (a+b) 1 = a+b. In this expansion, the m th term has powers a^{m}b^{n-m}. For example, if a binomial is raised to the power of 3, then looking at the 3rd row of Pascal’s triangle, the coefficients are 1, 3, 3 and 1. Mar 24, 2021 · Theorem \(\PageIndex{1}\) (Binomial Theorem) Pascal's Triangle; Summary and Review; Exercises ; A binomial is a polynomial with exactly two terms. Keep in mind that Sep 1, 2015 · Answer link. These coefficients of terms are derived from the triangle's first Oct 3, 2022 · Theorem 9. When a binomial expression is ra The following interactive lets you expand your own binomial expressions. Jan 17, 2016 · Then we can account for the factor of 2 of the 2x term, by multiplying by a sequence of powers of 2: 1,2,4,8,16,32,64. Added Feb 17, 2015 by MathsPHP in Mathematics. The rows are conventionally enumerated starting with row [latex]n=0[/latex] at the top, and the entries in each row are numbered from the left beginning with [latex]k=0[/latex]. MathAndScience. x² + 2xy + y². It's being used by the binomial expansion to find the corresponding coefficients for every row which represents the number increased by the extension term. Use the appropriate row of Pascal's triangle and a sequence of powers of 2 to find: (1+2x)^6 = 1 + 12x + 60x^2 + 160x^3 + 240x^4 Explanation: For any value of n, the nth power of a binomial is given by: (x +y)n = xn + nxn−1y + n(n − 1) 2 xn−2y2 + … +yn. khanacademy. General Binomial Expansion Formula. Dividing the exponent by 4 and having a remainder of 1 or 0 means the tens digit will be 0. To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form. The first element in any row of Pascal’s triangle is 1. May 5, 2020 · More Lessons: http://www. Use the binomial theorem to express ( x + y) 7 in expanded form. In this application, Pascal’s triangle will generate the leading coefficient of each term of a binomial Pascal's triangle and binomial expansion (Opens a modal) Expanding binomials Terms of binomial expansion. mr. Answer . Generating binomial coefficients in a row of Pascal's triangle from extensions of powers of eleven. The triangle is symmetrical. For the term involving we will need the term involving and the 3rd number in the 5th row from Pascal's triangle. In this video, I'm going to attempt to give you an intuition behind why multiplying binomials involve combinatorics Why we actually have the binomial coefficients in there at all. 5th row of Pascal's triangle is. This is a diagram of the coefficients of the expansion. To determine the expansion on ( x + y) 5, ( x + y) 5, we see n = 5, n = 5, thus, there will be 5+1 = 6 terms. The full revision guide is here https://www. For any binomial expansion of (a+b) n, the coefficients for each term in the expansion are given by the nth row of Pascal’s triangle. In algebra, the algebraic expansion of powers of a binomial is expressed by binomial expansion. Oct 25, 2015 · So, when expanding the power of a binomial, you must count how many possible combinations you have to find numbers i and j such that i+j=n. Binomial Theorem. A binomial expression is an algebraic expression with two terms. Pascal's triangle is a handy tool to quickly verify if the binomial expansion of the given polynomial is done correctly or not. Our pattern here is 0, 4, 4, 0. Just to give you an intuition. The Binomial Expansion formula for positive integer exponents. It explains this by saying that the relation $\\binom{n+1}{k} = \\binom{n}{k Binomial expansion Pascal's triangle can be used to identify the coefficients when expanding a binomial. Prior to the discussion of binomial expansion, this chapter will present Pascal's Triangle. Now on to the binomial. to get: 1,12,60,160,240,192,64. The variables m and n do not have numerical coefficients. 3: Polynomial Expansion and Pascal's Triangle is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. The first term and last term of the expansion are an a n and bn b n, respectively. 3. Notice the following pattern: In general, the k th term of any binomial expansion can be expressed as follows: Example 2. This chapter deals with binomial expansion; that is, with writing expressions of the form (a + b) [ n ] as the sum of several monomials. For example, to expand (x − 1)6 we would need two more rows of Pascal’s triangle, Oct 1, 2022 · Pascal’s triangle is an array of numbers forming a triangle that gives the coefficients of a binomial expansion and has many other interesting properties. (x + y) 3. Mar 26, 2014 · Practice this lesson yourself on KhanAcademy. The sum of the exponents for every term in the expansion is 2. I Answer link. This sequence is known as Pascal's triangle. Jan 2, 2022 · Learn how to use binomial theorem to expand binomials. The coefficients will correspond with line n+1 n + 1 of the triangle. The colors will actually be non-arbitrary this time. 6. . ( x + y) n. In the 3 rd row, flank the ends of the rows with 1 ′ s, and add 1 + 1 to find the middle number, 2. Binomial Theorem Calculator. Write out Pascal's triangle as far as the row that begins 1, 6 These are the coefficients you need for the expansion: (x+y)^6 = x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6 Why does this work? Pascal’s Triangle Examples. 1993 divided by 4 yields a remainder of 1. 1 A binomial expression is the sum, or difference, of two terms. How do I find the constant term of a binomial expansion? The expansion of a binomial is given by the Binomial Theorem: (x +y)n = ( n 0) ⋅ xn + (n 1) ⋅ xn−1 ⋅ y1 + +( n k) ⋅ xn−k ⋅ yk + + (n n) ⋅ yn = n ∑ k=0 ⋅ ( n k) ⋅ xn−k ⋅ yk. Apr 26, 2017 · MIT grad shows how to do a binomial expansion with the Binomial Theorem and/or Pascal's Triangle. The general formula for the expansion is: (x +y)n = n ∑ k=0 n! (n −k)!k! xn−kyk. (example in red) 1 1. 4. There are n + 1 n + 1 terms in the expansion. x 3 + 3x 2 Y + 3xY 2 + y 3. In the first line of each expansion, you'll see the numbers from Pascal's Triangle written within square brackets, [ ]. 4 questions. Pascal's Triangle & the Binomial Theorem 1. For example, the central number in the row is 20. How do we expand a product of polynomials? A polynomial with two terms is called a binomial. (2x+3)^3 = 8x^3 + 36x^2 + 54x + 27 With the Pascal's triangle, it's easy to find every binomial expansion : Each term, of this triangle, is the result of the sum of two terms on the top-line. We will now see how useful the triangle can be when we want to expand a binomial expression. Dividing the exponent by 4 and having a remainder of 2 or 3 means the tens digit will be 4. Hence: (1 +2x)6 = 1 + 12x +60x2 +160x3 + 240x4 +192x5 + 64x6. The coefficients are given by the eleventh row of Pascal’s triangle, which is the row we label 𝑛 = 1 0. 6 days ago · The binomial theorem was known for the case by Euclid around 300 BC, and stated in its modern form by Pascal in a posthumous pamphlet published in 1665. The exponent of b b increases by 1 1, from 0 0 to n n. Pascal's pamphlet, together with his correspondence on the subject with Fermat beginning in 1654 (and published in 1679) is the basis for naming the arithmetical triangle in his honor. For (a+b)6 ( a + b) 6, n = 6 n = 6 so the coefficients of the expansion will correspond with line 7 7. 2. Jun 16, 2017 · The coefficients of an extended binomial of the form are given as Pascal's Triangle, where n is the row of the rectangle. The binomial theorem gives a formula for expanding \((x+y)^n\) for any positive integer \(n\). The second line of each expansion is the result after tidying up. Jan 26, 2021 · 1. But with the Binomial theorem, the process is relatively fast! Expanding a binomial with a high exponent such as[latex]\,{\left(x+2y\right)}^{16}\,[/latex]can be a lengthy process. This step-by-step help will help yo Feb 19, 2024 · Notice that when we expanded (p + q) 4 (p + q) 4 in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal’s Triangle. The triangle is a simply an expression, or representation, of the following rule: starting at 1, make every number in the next the sum of the two numbers directly above it. Heliyon. Pascal's triangle and a calculator method are explained in detail. Visualisation of binomial expansion up to the 4th power In mathematics , the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem . 👉 Learn how to expand a binomial using binomial expansion. Pascal's Triangle is a triangle in which each row has one more entry than the preceding row Free Binomial Expansion Calculator - Expand binomials using the binomial expansion method step-by-step using pascals triangle. Type final answers in the spaces indicated. A diagram showing the first eight rows of Pascal's triangle. Pascal’s triangle always starts counting from 0, so to solve 8C6 (8 choose 6) we simply count 8 rows down, then 6 across. In binomial expansion, a polynomial (x + y)n is expanded into a sum involving terms of the form a x + b y + c. The binomial theorem describes the algebraic expansion of powers of a binomial. A very simple and practical way to expand binomials is to use a diagram called Pascal’s Triangle. According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending May 9, 2022 · We can see these coefficients in an array known as Pascal's Triangle, shown in Figure 13. and. The sum of the exponents of a a and b b in any term is n n. It is an easy way to determine the coefficients of binomial expansion. Consider the binomial expression a + b, and suppose we wish to find (a + b)2. For (a+b)10 ( a + b) 10, n = 10 n = 10 so the coefficients of the expansion will correspond with line 11 11. Our mission is to provide a free . The Binomial Series. May 8, 2013 · 👉 Learn how to expand a binomial using binomial expansion. Sep 17, 2023 · Binomial expansion; Probability; Combinatorics; In the binomial expansion of (x + y) n, the coefficients of each term are the same as the elements of the n th row in Pascal's triangle. Question 1. This section looks at Binomial Theorem and Pascals Triangle. Figure 13. Use notes/examplesfrom class. The sign of the 2nd term is negative in the 3rd example, as it should be. The theorem and its generalizations can be used to prove results and solve problems in combinatorics, algebra, calculus, and many Introduction and Summary. This is derived from 10 + 10 A video revising the techniques and strategies required for all of the AS Level Pure Mathematics chapter on Binomial Expansion that you need to achieve a gra 3 days ago · Binomial Theorem is a theorem that is used to find the expansion of algebraic identity (ax + by)n. The Binomial theorem tells us how to expand expressions of the form (a+b)ⁿ, for example, (x+y)⁷. org/math/algebra2/polynomial_and_rational/binomial_theorem/e/binomial-the Pascal triangle is the same thing. (x + y)². We do not need to fully expand a binomial to find a single specific term. The General Binomial Expansion ( n ≥ 1) This is a way of finding all the terms of the series, the coefficients and the powers of the variables. vu ln xe ek jo qf zt mn dg wc