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Chapter 1: Q.1.5 (page 17)

Determine the number of vectors (x1,...,xn),such that each xiis either 0or1and鈭�i=1nxi鈮�k.

Short Answer

Expert verified

There are 鈭�m=knn!m!(n-m)!different ways.

Step by step solution

01

Given Information.

The number of vectors x1,鈥�,xnis either 0 or 1and鈭�i=1nxi鈮�k.

02

Explanation.

The condition 鈭�i=1nxi鈮�kholds if and only if the vector x=x1,鈥�,xnhas kor more components equal to1.

Let us fix that the vector shall have exactly mcomponents equal to1. Therefore, we can arrange this collection of ones and zeros in n!different ways. But, because we do not want to count twice the permutations of zeros and ones among ourselves, we should divide byrole="math" localid="1648738905041" m!(m-k)!

The result follows by considering all mbetween kandn.

03

Explanation.

There are 鈭�m=knn!m!(n-m)!different ways.

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