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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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Most popular questions from this chapter

From a group of npeople, suppose that we want to choose a committee of k, k鈮�n, one of whom is to be designated as chairperson.

(a) By focusing first on the choice of the committee and then on the choice of the chair, argue that there are role="math" localid="1647945358534" nkkpossible choices.

(b) By focusing first on the choice of the non-chair committee members and then on the choice of the chair, argue that there are role="math" localid="1647945372759" nk-1n-k+1possible choices.

(c) By focusing first on the choice of the chair and then on the choice of the other committee members, argue that

there are role="math" localid="1647945385288" nn-1k-1possible choices.

(d) Conclude from parts (a), (b), and (c) that role="math" localid="1647945400273" knk=n-k+1nk-1=nn-1k-1.

(e) Use the factorial definition ofmr to verify the identity in part (d).

Give a combinatorial explanation of the identity

nr=nn-r

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