Prove Theorem 9.2.3 by mathematical induction.

To prove Theorem by mathematical induction.

“If n and r are integers and 1 ; ≤ ; r ; ≤ ; n then the number of r permutations of a set of n elements is given by the formula. P(n,r)=n(n−1)(n−2)......(n−r+1) first version or, equivalently,

P(n,r)=n!(n−r)! second version.”

Given information:

Theorem: If n and r are integers and 1 ; ≤ ; r ; ≤ ; n then the number of r permutations of a set of n elements is given by the formula. P(n,r)=n(n−1)(n−2)......(n−r+1).

Concept used:

Principle of mathematical induction is used to prove the mathematical statement.

Proof:

Theorem: If n and r are integers and 1 ; ≤ ; r ; ≤ ; n then the number of r permutations of a set of n elements is given by the formula. P(n,r)=n(n−1)(n−2)......(n−r+1).

Inductions on r :

r=1

P(n,1) Is the number of ways to choose orderly one element from n elements which is n.

Therefore, the statement is true for r=1.

Suppose the statement is true for r=k.

Therefore P(n,r)=n(n−1)(n−2)......(n−r+1).

First we select orderly k elements and P(n,k) is the number of ways to ordered selection of k elements taken from n elements