Binomial Theorem

10 Key excerpts on "Binomial Theorem"

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  Series and Products in the Development of Mathematics: Volume 1

  • Ranjan Roy(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 The Binomial Theorem 4.1 Preliminary Remarks The discovery of the Binomial Theorem for general exponents exerted a tremendous impact on the development of analysis, especially the theory of power series. It also led to an understanding that an exponential function was defined by the property f (a + b) = f (a)f (b). The Binomial Theorem was pivotal not only in the initial discovery of series for other important functions but also in the eventual consolidation of the foundations of analysis as a whole. The development of the theorem is particularly fascinating because it was independently found by both Newton and Gregory; because of the various approaches to its proof, including one by Euler; and because the validation of these proofs elicited the efforts of the best mathematicians of the nineteenth century. The Binomial Theorem for a positive integer exponent n states that (a + b) n = a n + A n 1 a n−1 b + A n 2 a n−2 b 2 + · · · + A n n−1 ab n−1 + b n , (4.1) where the coefficients A n k satisfy the additive rule A n k = A n−1 k−1 + A n−1 k , (4.2) and the multiplicative rule A n k = n(n − 1) · · · (n − k + 1) 1 · 2 · · · k , (4.3) where it is understood that A n 0 = 1. We here use a notation unusual today, because the notation  n k  or C n k , or C n,k , may be suggestive of recent developments, whereas we wish to understand how these coefficients developed over time. Now we note that the additive rule (4.2) is not difficult to obtain. In terms of the notation used in (4.1), we can write 77 78 The Binomial Theorem (a + b) n−1 = a n−1 + A n−1 1 a n−2 b + · · · + A n−1 k−1 a n−k b k−1 + A n−1 k a n−k−1 b k + · · · + b n−1 . Multiplying both sides by a + b, we have (a + b) ( a n−1 + · · · + A n−1 k−1 a n−k b k−1 + A n−1 k a n−k−1 b k + · · · + b n−1 ) = a n + · · · + A n k a n−k b k + · · · + b n . Equating the coefficients of a n−k b k on each side, we obtain (4.2). The multiplicative rule is somewhat more difficult to obtain.

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  Sources in the Development of Mathematics

  Series and Products from the Fifteenth to the Twenty-first Century

  • Ranjan Roy(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 The Binomial Theorem 4.1 Preliminary Remarks The discovery of the Binomial Theorem for general exponents exerted a tremendous impact on the development of analysis, especially the theory of power series. It also led to an understanding that an exponential function was defined by the property f (a + b) = f (a)f (b). The Binomial Theorem was pivotal not only in the initial dis- covery of series for other important functions but also in the eventual consolidation of the foundations of analysis as a whole. The development of the theorem is particularly fascinating because it was independently found by both Newton and Gregory; because of the various approaches to its proof, including one by Euler; and because the vali- dation of these proofs elicited the efforts of the best mathematicians of the nineteenth century. Islamic mathematicians were the original discoverers of the Binomial Theorem for positive integral exponents, although they did not have the notation to write the expan- sion for arbitrary integers. But they knew how to find the coefficients for any given integral exponent. The two important rules for binomial coefficients appear in the work of al-Kashi of around 1427, and it is likely that earlier Islamic mathematicians such as al-Tusi, Omar Khayyam, and al-Karji were also aware of them: Let C n,k denote the coefficient of x k in the expansion of (1 + x) n . Then C n,k = C n−1,k + C n−1,k−1 and C n,k = n(n − 1) ··· (n − k + 1) k! . The first formula, the additive rule for binomial coefficients, leads to the expansion of (1 + x) n , by means of the expansion of (1 + x) n−1 ; the second formula, the multiplicative rule, immediately yields the expansion of (1 + x) n . Henry Briggs (1561–1630) appears to be the first European to explicitly state both formulas, though Cardano may have known the results around 1570. In 1654, Pascal gave a proof by complete induction of the second formula.

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  Combinatorics

  • Nicholas Loehr(Author)
  • 2017(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  To see why, rewrite the given identity as (p - q) (r) = 0, where p - q is a polynomial in the formal variable r. If p - q were nonzero, say of degree d, then p - q would have at most d real roots. Since we are assuming the equation holds for infinitely many values of r, we conclude that p - q is the zero polynomial. Thus, p (r) = q (r) must hold for every r. A similar result holds for polynomial identities involving multiple parameters. 2.3 The Binomial Theorem The Binomial Theorem is a famous formula for expanding the n th power of a sum of two terms. Binomial coefficients are so named because of their appearance in this theorem. 2.8. The Binomial Theorem. For all x, y ∈ R and all n ∈ Z ≥ 0, (x + y) n = ∑ k = 0 n n k x k y n - k. For example, (x + y) 4 = 1 y 4 + 4 x y 3 + 6 x 2 y 2 + 4 x 3 y + 1 x 4. We now give a combinatorial proof of the Binomial Theorem under the additional assumption that x and y are positive integers. (Since both sides of the formula are polynomials in x and y, we can deduce the general case using the result mentioned in Remark 2.7.) Let A = { V 1, …, V x, C 1, …, C y } be an alphabet consisting of x + y letters, where A consists of x vowels V 1, …, V x and y consonants C 1, …, C y. Let S be the set of all n -letter words using the alphabet A. By the Word Rule, | S | = (x + y) n. On the other hand, we can classify words in S based on how many vowels they contain. For 0 ≤ k ≤ n, let S k be the set of words in S that contain exactly k vowels (and hence n - k consonants). To build a word in S k, first choose a set of k positions out of n where the vowels will appear (n k ways, by the Subset Rule); then fill these positions with a sequence of vowels (x k ways, by the Word Rule); then fill the remaining positions with a sequence of consonants (y n - k ways, by the Word Rule). By the Product Rule, | S k | = n k x k y n - k

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  Mathematical Foundations of Computer Science

  • Bhavanari Satyanarayana, T.V. Pradeep Kumar, Shaik Mohiddin Shaw(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  C HAPTER - 14 Binomial Theorem LEARNING OBJECTIVES ♦ to understand the basic concepts of Binomial and Multinomial Coefficients ♦ to know some Generating functions of permutations and combinations ♦ to develop problem solving skills related to the Principle of Inclusion & Exclusion If x and a are real numbers then for all n ∈ N we have. that (x+a) n = n C 0 x n a 0 + n C 1 x n − 1 a 1 + n C 2 x n − 2 a 2 + … + n C r x n − r a r + … + n C n − 1 x 1 a n − 1 + n C n x 0 a n That is, (x +. a) n = ∑ r = 0 n C n r x n − r a r and it is called as Binomial Theorem. 14.1  Binomial and Multinomial Coefficients 14.1.1  Pascal’s Triangle We know that (x + a) 0 =. 1 (x+a) 1 = x + a = 1. x + 1. a (x+a) 2 = x 2 + 2 ax+a 2 = 1. x 2 + 2. ax+1.a 2 (x+a) 3 = x 3 + 3 x 2 a + 3 xa 2 + a 3 = 1. x 3 + 3. x 2 a+3.xa 2 + 1. a 3 We observe that the coefficients in the above expansions follow a particular pattern as shown in the Table 14.1. Table 14.1 We observe that each row is bounded by 1 on both sides. Any entry, except the first and last, in a row is the sum of two entries in the preceding row, one on the immediate left and the other on immediate right. The above pattern is known as Pascal’s triangle. It has been checked that the above pattern also holds good for the coefficients in the expansions of the binomial expressions having index (exponent) greater than 3 as shown in the Table 14.2. Table 14.2 Index of the binomial Coefficients of various terms 0 1 1 1 1 2 1  2  1 3 1  3  3  1 4 1   4  6  4   1 5 1  5  10  10  5  1 Using the above Pascal’s triangle, we may express (x + a) n for n = 1, 2, 3,. … (x + a) 1 = x + a (x+a) 2 = x 2 + 2 ax + a 2 (x + a) 3 = x 3 + 3 ax 2 + 3 a 2 x + a 3 (x + a) 4 = x 4 + 4 ax 3 + 6 a 2 x 2 + 4 a 3 x + a 4 (x+a) 5 = x 5 + 5 ax 4 + 10 a 2 x 3 + 10 a 3 x[-. -=PLGO-SEPARATOR=--]2 + 5a 4 x + a 5 and so on. 14.1.2  Properties of Binomial Coefficients or Combinatorial Identities An identity that results from some counting process is called a combinatorial identity

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  Discrete Mathematics with Ducks

  • sarah-marie belcastro, Sarah-Marie Belcastro(Authors)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Then evaluate it, somehow. 6.7 Binomial Basics Hey! You! Don’t read this unless you have worked through the problems in Section 1.6. I mean it! A binomial is a polynomial with exactly two terms, such as 3 a - 2 b 5. Consider the simple binomial (x + y). We can rewrite (x + y) n as (x + y) · (x + y) · (x + y) · ⋯ · (x + y), where there are n copies of (x + y) in that product. If we expand thebinomial power (x + y) n into a polynomial, we know what all of the variable parts of the terms are—they’re x n, x n - 1 y, x n - 2 y 2, ⋯, x 2 y n - 2, x y n - 1, y n. But we don’tknow what the coefficients are. Well, we know they should be called binomial coefficients because they are coefficients of a binomial expansion, but that’s not yet helpful. Notice that each term has variables whose degrees total to n. (Yes, literally go back a few sentences and notice it actively.) We will have one occurrence of a given term in the full expansion for every way there is of forming it, meaning, for example, that there will be only one copy of x n because we can only form x n by multiplying together all n copies of x in (x + y) · (x + y) · (x + y) · ⋯ · (x + y). We can form x n - 1 y by multiplying together all but one of the x s and the remaining y, and we can figure out the number of different x n - 1 y s we can have by counting the number of ways of choosing that single y. Oh, hey! That’s n 1, and in fact, the number of ways of forming x n was n 0. Similarly, the number of ways to form x n - k y k is n k because we choose k copies of y from the n copies of (x + y) (or, equivalently, because we choose n - k copies of x from the n copies of (x + y)). Yup. Conclusion 1: Another name for n k is binomial coefficient. Conclusion 2: We have proven a theorem. Theorem 1.6 [the Binomial Theorem] (x + y) n = ∑ k = 0 n n k x n - k y k = n 0 x n + n 1 x n - 1 y + ⋯ + n n - 1 x y n - 1 + n n y n. This theorem has lots of cool consequences

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  Principles and Techniques in Combinatorics

  • Chen Chuan-Chong, Koh Khee-Meng(Authors)
  • 1992(Publication Date)
  • WSPC

    (Publisher)

  B|, as claimed.

  We thus conclude that the coefficient of (2.8.6) in the expansion is given by

  Combining this with identity (2.8.5), we arrive at the following generalization of the Binomial Theorem, that was first formulated by G.W. Leibniz (1646-1716) and later on proved by Johann Bernoulli (1667-1748).

  Theorem 2.8.1 (The Multinomial Theorem). For n, m ∈ N,

  where the sum is taken over all m-ary sequences (n1 , n2 , …, n

  m

  ) of nonnegative integers with , and

  Example 2.8.1. For n = 4 and m = 3, we have by Theorem 2.8.1 ,

  Because of Theorem 2.8.1 , the numbers of the form (2.8.2) are usually called the multinomial coefficients. Since multinomial coefficients are generalizations of binomial coefficients, it is natural to ask whether some results about binomial coefficients can be generalized to multinomial coefficients. We end this chapter with a short discussion on this.

  1° The identity for binomial coefficients may be written as (here of course n1 + n2 = n). By identity (2.8.5), it is easy to see in general that

  where {α(l), α(2), …, α(m)} = {1, 2 m}.

  2° The identity for binomial coefficients may be written:

  In general, we have:

  3° For binomial coefficients, we have the identity By letting x1 = x2 = ··· = x

  m

  = 1 in the multinomial theorem, we have

  where the sum is taken over all m-ary sequences (n1 , n2 , …, n

  m

  ) of nonnegative integers with .

  Identity (2.8.9) simply says that the sum of the coefficients in the expansion of (x1 + x2 +…+ x

  m

  )

  n

  is given by m

  n

  . Thus, in Example 2.8.1 , the sum of the coefficients in the expansion of (x1 + x2 + x3 )4 is 81, which is 34 .

  4° In the binomial expansion the number of distinct terms is n + 1. How many distinct terms are there in the expansion of (x1 + x2 +…+ x

  m

  )

  n

  ? To answer this question, let us first look at Example 2.8.1 . The distinct terms obtained in the expansion of (x1 + x2 + x3 )4

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  Maths: A Student's Survival Guide

  A Self-Help Workbook for Science and Engineering Students

  • Jenny Olive(Author)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  7 Binomial series and proof by induction In this chapter we find out how to do binomial expansions, and see how they can describe some real-life situations. We also look at a new method of proving mathematical statements. The chapter is divided into the following sections. 7.A Binomial series for positive whole numbers (a) Looking for the patterns, (b) Permutations or arrangements, (c) Combinations or selections, (d) How selections give binomial expansions, (e) Writing down rules for binomial expansions, (f) Linking Pascal’s Triangle to selections, (g) Some more binomial examples 7.B Some applications of binomial series and selections (a) Tossing coins and throwing dice, (b) What do the probabilities we have found mean? (c) When is a game fair? (Or are you fair game?) (d) Lotteries: winning the jackpot . . . or not 7.C Binomial expansions when n is not a positive whole number (a) Can we expand (1 + x ) n if n is negative or a fraction? If so, when? (b) Working out some expansions, (c) Dealing with slightly different situations 7.D Mathematical induction (a) Truth from patterns – or false mirages? (b) Proving the Binomial Theorem by induction, (c) Two non-series applications of induction 7.A Binomial series for positive whole numbers 7.A. (a) Looking for the patterns The first half of this chapter describes what are called binomial series. I have given them so much space because they have many applications. For this reason it is important that you should be able to do binomial expansions correctly and happily. The word ‘binomial’ comes from the two quantities put together in a bracket which we start from. Binomial expansions are what we get when we raise these brackets to different powers and then multiply the brackets together to find the result. In this first section all these powers will be positive whole numbers. Here are some examples. ( a + b ) 1 is just a + b ( a + b ) 2 = ( a + b )( a + b ) = a 2 + 2 ab + b 2 .

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  Thinking About Equations

  A Practical Guide for Developing Mathematical Intuition in the Physical Sciences and Engineering

  • Matt A. Bernstein, William A. Friedman(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Equation 7.1 is certainly true, but it is not very general and is of limited use. A more general expression is

  (7.2)     

  where x and y are real variables. Equation 7.2 covers more cases and reduces to Equation 7.1 when x = 3 and y = 5. We can continue to generalize Equation 7.2 by allowing the exponent to be any positive integer n . This yields the binomial formula, which is a finite series that terminates after n + 1 terms:

  (7.3)     

  When n = 2, Equation 7.3 contains three terms and reduces to Equation 7.2 .

  Continuing to generalize Equation 7.3 , consider the case where the exponent is an arbitrary real number r . Provided that y ≠ 0, let u = x /y , yielding

  (7.4)    

  If we expand (1 + u )r into an infinite sum using the Taylor series expansion about u = 0, we obtain

  (7.5)     

  To ensure that the series in Equation 7.5 converges, we can choose y to be the element of the pair (x , y ) that has the larger absolute value, which implies that |u | < 1. Because a generalized equation covers more cases, side conditions, such as |u | < 1, frequently need to be specified to ensure its validity. The binomial formula can be generalized further in several different ways. Exercise 7.6 discusses one particular generalization known as the multinomial formula.

  The progression from Equation 7.1 to Equation 7.5 illustrates that although more general equations apply to a wider range of situations, they also tend to be more abstract. To better understand the meaning of the general expression, it is sometimes helpful to reduce it to one of the embedded special cases that is more familiar or concrete, e.g., to go from Equation 7.3 to Equation 7.2 by substituting n = 2.

  Not only do more general expressions tend to be more abstract, but they also tend to be more complicated or messy as well. This again is illustrated by considering the progression from Equation 7.2 to Equation 7.5 . Sometimes, the use of compact notation can reduce (or at least hide) the apparent complexity. The standard definition of the binomial coefficient “n choose k ” is

  (7.6)     

  with 0! = 1, by definition. (Binomial coefficients were used in Example 4.1 .) Then, we can rewrite the binomial formula of Equation 7.3 more compactly by using the binomial coefficients defined in Equation 7.3

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  Precalculus

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Example 5 Exercises 10.6 863 Skills In Exercises 1−4, write down the variable parts of the terms in the expansion of the binomial. 1. ( a + b ) 5 2. ( a + b ) 6 3. ( x + y ) 7 4. ( x + y ) 8 In Exercises 5−16, evaluate each expression. 5. 4! 6. 6! 7. 3! __ 2! 8. 4! __ 3! 9. ( 6 2 ) 10. ( 5 3 ) 11. ( 7 5 ) 12. ( 7 4 ) 13. ( 10 10 ) 14. ( 10 0 ) 15. ( 100 0 ) 16. ( 100 100 ) In Exercises 17−28, use the Binomial Theorem to expand the expression. 17. ( x + 2) 4 18. ( x − 3) 3 19. (2 x − l) 3 20. (2 x + 3) 4 21. (3 + y ) 5 22. (4 − z ) 4 23. ( x − 3 z ) 4 24. (2 z + y ) 3 25. ( x 2 + l) 3 26. ( x 2 − 2) 3 27. ( y − 2 x ) 4 28. ( z + 4 x ) 5 In Exercises 29−42, use the Binomial Theorem to find the indicated term or coefficient. 29. The coefficient of x 3 when expanding ( x + 4) 5 30. The coefficient of y 2 when expanding ( y − 3) 5 31. The coefficient of x 5 when expanding (3 x + 2) 6 32. The coefficient of y 4 when expanding (2 y + 1) 7 33. The coefficient of x 6 when expanding ( x + 1 ) 8 34. The coefficient of y 7 when expanding ( y − 3) 10 35. The third term in the expansion of ( x − 4) 6 36. The fourth term in the expansion of ( x + 3) 6 37. The sixth term in the expansion of ( x + 4 y ) 5 38. The seventh term in the expansion of ( a + 2 b ) 6 39. The fifth term in the expansion of (3 x − 2) 6 40. The fifth term in the expansion of (3 x + l) 8 41. The fourth term in the expansion of (4 x − 2) 6 42. The fourth term in the expansion of (3 x − l) 8 Concepts 43. Show that ( n r ) = ( n n − r ) , where 0 ≤ r ≤ n , with n and r integers. 44. Show that ( n 0 ) = 1. 45. Evaluate the following. ( 4 0 ) ( 1 __ 3 ) 4 + ( 4 1 ) ( 1 __ 3 ) 3 ( 2 __ 3 ) + ( 4 2 ) ( 1 __ 3 ) 2 ( 2 __ 3 ) 2 + ( 4 3 ) ( 1 __ 3 )( 2 __ 3 ) 3 + ( 4 4 ) ( 2 __ 3 ) 4 10.7 10.7 Mathematical Induction 865 Mathematical Induction Many mathematical facts are established by first observing a pattern, then making a conjecture about the general nature of the pattern, and finally proving the conjecture.

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  An Invitation to Combinatorics

  • Shahriar Shahriari(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  5 Binomial and Multinomial Coefficients Let n be a non-negative integer. The n + 1 binomial coefficients ( n 0 ) , ( n 1 ) , . . . , ( n n ) , respectively, count the number of subsets of size 0, 1, . . . , n of a set with n elements. As a result, the average of these numbers is 2 n /(n + 1) and often not an integer. But the average of the squares of these numbers is always an integer. Why? 5.1 Binomial Coefficients Let n be a non-negative integer. Recall that, for n > 0, [n] = {1, 2, . . . , n}, [0] = ∅, and for 0 ≤ k ≤ n, the binomial coefficient ( n k ) is defined to be the number of subsets of size k of [n]. We showed in Theorem 4.21c that  n k  = n! k! (n − k)! . The binomial coefficients are a set of numbers that come up often. They appear in many counting problems and they satisfy many identities. In this chapter, we explore some of their properties. Since we have a formula for these numbers, we can certainly use the formula to prove identities involving the binomial coefficients. However, we can also use induction—often using the identity of Theorem 5.2c below—as well as “combinatorial arguments” and the “Binomial Theorem.” The latter will be treated in Section 5.2. 5.1.1 Combinatorial Proofs and Binomial Coefficient Identities Warm-Up 5.1. Forty-seven individuals are gathered in a room. You first choose 7 from the group of 47, and then 4 from the 7, and give each of the 4 a blue scarf. In how many ways can you do that? What if you pick 4 from the group of 47, give each a blue scarf, and then choose 3 more people from the remaining 43? Are the two answers the same? Why or why not? If possible, generalize and give an identity. The next theorem gives examples of combinatorial proofs of counting identities. In such proofs we either show that we can count the same set of objects in two different ways and, as a result, get an identity, or we find a bijection between two sets of objects showing that the two sets have equal numbers of elements. 147

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