# MATH239 Study Guide - Midterm Guide: Local Exchange Trading System, Opata Language, 5,6,7,8

## Document Summary

Question 1. (a) explain (either with the binomial theorem or with a combinatorial argument) why, for any non-negative integer k, (cid:19) (cid:18)k k(cid:88) (cid:1)xk. Solution: by the binomial theorem, (1 + x)k =(cid:80)k (cid:18)k (cid:19) k(cid:88) 2k = (1 + 1)k = i i=0 as desired. Alternatively, the left side is the number of subsets of a k-set, while the right side gives the sum of the numbers of subsets of a k-set of the di erent sizes i = 0, 1, 2 . Using this fact and (a) or otherwise, show that (cid:19) (cid:18)k k(cid:88) i k! i! (k i)! 2k = (cid:18)k (cid:19) i k(cid:88) k(cid:88) i=0 k! i! (k i)! i=0 k(cid:88) i=0. Solution: we just proved that i=0 and, for each i, we know. Dividing by k! throughout, we get (cid:18)k i (cid:19) 2k k! as desired. (c) consider the power series (cid:88) n 0 xn n! p(x) =