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mdm12Section5_4_OddSolutionsFinal

mdm12Section5_4_OddSolutionsFinal - Q1-5 Q11 Q7 Q13-17 Q9...

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Q1-5 Q7 Q9 Q11 Q13-17 Q19-21 CHAPTER 5 Section 5.4, pp. 293–295 Solutions to Odd Number Problems 1. a) 17 C 11 = 16 C 10 + 16 C 11 b) 43 C 36 = 42 C 35 + 42 C 36 c) n +1 C r +1 = n C r + n C r +1 d) 32 C 4 + 32 C 5 = 33 C 5 e) 15 C 10 + 15 C 9 = 16 C 10 f) n C r + n C r +1 = n +1 C r +1 g) 18 C 9 17 C 9 = 17 C 9 + 17 C 8 17 C 9 = 17 C 8 h) 24 C 8 23 C 7 = 23 C 8 + 23 C 7 23 C 7 = 23 C 8 i) n C r n -1 C r -1 = n -1 C r + n -1 C r -1 n -1 C r -1 = n -1 C r 3. For a binomial expansion with exponent n , there are n + 1 terms. a) 13 b) 6 c) 21 5. a) 9 C 0 + 9 C 1 + + 9 C 9 = (1 + 1) 9 = 2 9 = 512 b) 12 C 0 12 C 1 + 12 C 2 + 12 C 11 + 12 C 12 = (1 – 1) 12 = 0 c) d) ( ) 15 15 15 0 15 1 1 2 32 768 r r C = = + = = ( ) 15 0 1 1 2 n n r r n C = = + = Top
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7. a) i) Consider (1 + x ) n ( x + 1) n = (1 + x ) 2 n . RHS = ( 2 n C 0 + 2 n C 1 x + n C 2 x 2 + ……+ 2 n C n x n + … + 2 n C 2 n x 2 n ) In the RHS, the coefficient of x n is 2 n C n . LHS = ( n C 0 + n C 1 x + n C 2 x 2 + ……+ n C n -1 x n -1 + n C n x n )( n C 0 x n + n C 1 x n -1 + n C 2 x n -2 + …+ n C n ) In the expansion of LHS, the coefficient of x n is C 2 ( n ,0) + C 2 ( n ,1) + C 2 ( n ,2) + … C 2 ( n , n ). The coefficients of x n must be equal. C 2 ( n ,0) + C 2 ( n ,1) + C 2 ( n ,2) + … C 2 ( n , n ) = C (2 n , n ) = 2 n C n ii) Consider (1 + x ) n (1 - x ) n = [(1 + x )(1 – x )] n = (1 – x 2 ) n LHS = ( n C 0 x n + n C 1 x n -1 + n C 2 x n -2 + ……+ n C n -1 x 1 + n C n )( n C 0 - n C 1 x 1 + n C 2 x 2 + … + (-1) n n C n x n ) In the expansion of LHS, the coefficient of x n is C 2 ( n ,0) - C 2 ( n ,1) + C 2 ( n ,2) + … (-1) n C 2 ( n , n ) RHS = n C 0 - n C 1 x 2 + n C 2 x 4 + …+ (-1) n n C n x 2 n In the RHS, there are only terms of x to the even exponents, therefore the coefficient of x n = ( ) /2 /2 0, when is odd 1
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