MATH 271 Fall 2010 Assignment 3 Solutions - 1 MATHEMATICS 271 L01 FALL 2010 ASSIGNMENT 3 SOLUTIONS 1 In this question n and k are integers For part(a

MATH 271 Fall 2010 Assignment 3 Solutions - 1 MATHEMATICS...

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1 MATHEMATICS 271 L01 FALL 2010 ASSIGNMENT 3 SOLUTIONS 1. In this question,nandkare integers. For part (a) and (b),n°k°0. For part (c)and (d),n°k°1.(a) Prove thatnPi=0°ni±= 2nusing the Binomial Theorem. (b) Prove thatnPi=0nby a combinatorial proof. (c) Prove thatk°nkk°1 (d) Prove thatnPk=1k°nkn°1.
2 n P k =1 k ° n k ± = n P k =1 n ° n ° 1 k ° 1 ± by part (c) we know that k ° n k ± = n ° n ° 1 k ° 1 ± = n n P k =1 ° n ° 1 k ° 1 ± = n n P k =1 ° n ° 1 k ° 1 ± = n n ° 1 P i =0 ° n ° 1 i ± by putting i = k ± 1 = n 2 n ° 1 : because n ° 1 P i =0 ° n ° 1 i ± = 2 n ° 1 by part (a)

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