Binomial
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The document discusses binomial coefficients and the binomial theorem. It provides examples of expanding binomial expressions using Pascal's triangle and the binomial coefficient formula. The key points are: 1) Binomial coefficients describe the coefficients that arise when expanding binomial expressions using the binomial theorem. 2) Pascal's triangle provides a visual representation of the binomial coefficients. 3) The binomial theorem can be used to expand binomial expressions in terms of binomial coefficients and write individual terms.
Binomial
- 1. The Binomial & Multinomial Coefficients
- 2. ? In formulas arising from the analysis of algorithms in computer science, the binomial coefficients occur over and over again, so that a facility for manipulating them is a necessity. ? Moreover, different approaches to problems often give rise to formulas that are different in appearance yet identities of binomial coefficients reveal that they are, in fact, the same expressions.
- 3. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 3 The binomial theorem provides a useful method for raising any binomial to a nonnegative integral power. Consider the patterns formed by expanding (x + y)n. (x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 Notice that each expansion has n + 1 terms. 1 term 2 terms 3 terms 4 terms 5 terms 6 terms Example: (x + y)10 will have 10 + 1, or 11 terms.
- 4. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 4 Consider the patterns formed by expanding (x + y)n. (x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 1. The exponents on x decrease from n to 0. The exponents on y increase from 0 to n. 2. Each term is of degree n. Example: The 5th term of (x + y)10 is a term with x6y4.¡±
- 5. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 5 The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry. The coefficient of xn¨Cryr in the expansion of (x + y)n is written or nCr . n r ? ? ? ? ? ? (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 The first and last coefficients are 1. The coefficients of the second and second to last terms are equal to n. 1 1 Example: What are the last 2 terms of (x + y)10 ? Since n = 10, the last two terms are 10xy9 + 1y10. So, the last two terms of (x + y)10 can be expressed as 10C9 xy9 + 10C10 y10 or as xy9 + y10. ?? ? ? ?? ? ? 9 10 ?? ? ? ?? ? ? 10 10
- 6. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 6 The triangular arrangement of numbers below is called Pascal¡¯s Triangle. Each number in the interior of the triangle is the sum of the two numbers immediately above it. The numbers in the nth row of Pascal¡¯s Triangle are the binomial coefficients for (x + y)n . 1 1 1st row 1 2 1 2nd row 1 3 3 1 3rd row 1 4 6 4 1 4th row 1 5 10 10 5 1 5th row 0th row1 6 + 4 = 10 1 + 2 = 3
- 7. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Use the fifth row of Pascal¡¯s Triangle to generate the sixth row and find the binomial coefficients , , 6C4 and 6C2 . 6 1 ? ? ? ? ? ? 6 5 ? ? ? ? ? ? 5th row 1 5 10 10 5 1 6th row 6 0 ? ? ? ? ? ? 6 1 ? ? ? ? ? ? 6 2 ? ? ? ? ? ? 6 3 ? ? ? ? ? ? 6 4 ? ? ? ? ? ? 6 5 ? ? ? ? ? ? 6 6 ? ? ? ? ? ? 6C0 6C1 6C2 6C3 6C4 6C5 6C6 = 6 = and 6C4 = 15 = 6C2. 6 1 ? ? ? ? ? ? 6 5 ? ? ? ? ? ? There is symmetry between binomial coefficients. nCr = nCn¨Cr 1 6 15 20 15 6 1
- 8. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Use Pascal¡¯s Triangle to expand (2a + b)4. (2a + b)4 = 1(2a)4 + 4(2a)3b + 6(2a)2b2 + 4(2a)b3 + 1b4 = 1(16a4) + 4(8a3)b + 6(4a2b2) + 4(2a)b3 + b4 = 16a4 + 32a3b + 24a2b2 + 8ab3 + b4 1 1 1st row 1 2 1 2nd row 1 3 3 1 3rd row 1 4 6 4 1 4th row 0th row1
- 9. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 9 The symbol n! (n factorial) denotes the product of the first n positive integers. 0! is defined to be 1. n! = n(n ¨C 1)(n ¨C 2) ? 3 ? 2 ? 1 1! = 1 4! = 4 ? 3 ? 2 ? 1 = 24 6! = 6 ? 5 ? 4 ? 3 ? 2 ? 1 = 720 Formula for Binomial Coefficients For all nonnegative integers n and r, ! ( )! ! n r n C n r r ? ? Example: )123()1234( )123()4567( ?????? ?????? ? !3!4 7 !3!4 !7 !3)!37( !7 ??? 37 ?? ? ?C 35 1234 4567 ??? ??? ??
- 10. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Use the formula to calculate the binomial coefficients 10C5, 10C0, and .50 48 ? ? ? ? ? ? 12 1 ? ? ? ? ? ? !5)!510( !10 ? 510 ? ?C !0)!010( !10 ? 010 ? ?C !48)!4850( !50 48 50 ?? ??? ? ? ?? ? ? !1)!112( !12 1 12 ?? ??? ? ? ?? ? ? !5!5 !10 ? ? !5!5 !5)678910( ? ????? ? 252 12345 678910 ???? ???? ?? !0!10 !10 ? ? 1 1 1 !0 !1 ??? !48!2 !48)4950( !48!2 !50 ? ?? ? ?? 1225 12 4950 ? ? ?? !1!11 !1112 !1!1 !12 ? ? ? ?? 12 1 12 ??
- 11. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 11 Binomial Theorem 1 1 ( )n n n n r r n n n rx y x nx y C x y nxy y? ? ? ? ? ? ? ? ? ? ?L L ! with ( )! ! n r n C n r r ? ?
- 13. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 13 Binomial Theorem Example: Use the Binomial Theorem to expand (x4 + 2)3. 03C 13C 23C 33C?? 34 )2(x ?34 )(x ?)2()( 24 x ?24 )2)((x 3 )2( 1 ?34 )(x? 3 ?)2()( 24 x 3 ?24 )2)((x 1 3 )2( 8126 4812 ???? xxx
- 14. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 14 Although the Binomial Theorem is stated for a binomial which is a sum of terms, it can also be used to expand a difference of terms. Simply rewrite (x + y)n as (x + (¨C y))n and apply the theorem to this sum. Example: Use the Binomial Theorem to expand (3x ¨C 4)4. 4 )43( ?x 4 ))4(3( ??? x 432234 )4(1)4)(3(4)4()3(6)4()3(4)3(1 ????????? xxxx 256)64)(3(4)16)(9(6)4)(27(481 234 ??????? xxxx 25676886443281 234 ????? xxxx
- 15. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 15 Example: Use the Binomial Theorem to write the first three terms in the expansion of (2a + b)12 . ...)2( 2 12 )2( 1 12 )2( 0 12 )2( 210111212 ??? ? ? ?? ? ? ??? ? ? ?? ? ? ??? ? ? ?? ? ? ?? babaaba ...)2( !2)!212( !12 )2( !1)!112( !12 )2( !0)!012( !12 2101011111212 ??? ? ? ? ? ? ? ? babaa ...)2)(1112()2(12)2( 2101011111212 ? ???? babaa ...135168245764096 2101112 ???? babaa
- 16. Copyright ? by Houghton Mifflin Company, Inc. All rights reserved. 16 Example: Find the eighth term in the expansion of (x + y)13 . Think of the first term of the expansion as x13y0 . The power of y is 1 less than the number of the term in the expansion. The eighth term is 13C7 x6y7. Therefore, the eighth term of (x + y)13 is 1716 x6y7. !7!6 !7)8910111213( !7!6 !13 ? ?????? ? 713 ??C 1716 123456 8910111213 ????? ????? ??
- 17. Find the 6th term in the expansion of (3a + 2b)12 Using the Binomial Theorem, let x = 3a and y = 2b and note that in the 6th term, the exponent of y is m = 5 and the exponent of x is n ¨C m = 12 ¨C 5 = 7. Consequently, the 6th term of the expansion is: ?57 512 yxC ? ? ? ?57 23 !5!7 !789101112 ba ? ????? = 55,427,328 a7b5
- 18. E.g. 7¡ªFinding a Particular Term in a Bin. Expansion Find the coefficient of x8 in the expansion of ? Both x2 and 1/x are powers of x. ? So, the power of x in each term of the expansion is determined by both terms of the binomial. 10 2 1 x x ? ? ?? ? ? ?
- 19. To find the required coefficient, we first find the general term in the expansion. ? By the formula, we have: a = x2, b = 1/x, n = 10 ? So, the general term is: E.g. 7¡ªFinding a Particular Term in a Bin. Expansion 10 2 2 1 10 3 10 10 101 ( ) ( ) 10 10 10 10 r r r r r x x x r x r x r ? ? ? ? ? ? ? ?? ? ?? ? ? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?
- 20. Thus, the term that contains x8 is the term in which 3r ¨C 10 = 8 r = 6 ? So, the required coefficient is: E.g. 7¡ªFinding a Particular Term in a Bin. Expansion 10 10 210 10 6 4 ? ? ? ? ? ?? ? ? ? ?? ? ? ?