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Chapter 1: Q. 1.6 (page 17)

How many vectors x1,...,xkare there for which each role="math" localid="1647853392605" xiis a positive integer such that role="math" localid="1647853435585" 1鈮�xi鈮�nandrole="math" localid="1647853511159" x1<x2<路路路<xk?

Short Answer

Expert verified

The number of vectors areCkn=n!k!(n-k)!.

Step by step solution

01

Step 1. Given information.

It is given that,

xiis a positive integer.

1鈮�xi鈮�n, it means all the numbers in the set lies in the range (1,n).

x1<x2<路路路<xk, it means the numbers should be ascending order. As out of n, k distinct integers are chosen so there can be only one way of arrangement.

02

Step 2. State the answer.

So, to get a set of numbers fulfilling the given conditions is same as selecting k numbers randomly from n numbers, which can be done in Ckn=n!k!(n-k)!ways.

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