7 2 Permutations – Class 11TH Math | Theory of Intelligence

7.2 Permutations

An arrangement of all or part of a set of objects in a specific order is a permutation.

{}^n P_r = P(n,r) = \dfrac{n!}{(n-r)!} \quad (r \le n)

Theorem. ${}^n P_r = n(n-1)\cdots(n-r+1) = \dfrac{n!}{(n-r)!}$ .

Proof. First place: $n$ choices; second: $n-1$ ; …; $r$ th: $n-r+1$ . Multiply by the fundamental principle.

Reason $0! = 1$ : when $n = r$ , ${}^n P_n = \dfrac{n!}{0!} = n!$ forces $0! = 1$ .

Different $4$ -digit numbers from $1,2,3,4,5,6$ (no repeat): ${}^6 P_4 = 360$ .

$4$ math, $3$ physics, $2$ chemistry books on a shelf of $9$ :

(a) No restriction: $9! = 362880$ .

(b) Same subjects together: treat $3$ subject blocks $\Rightarrow$ $4!\,3!\,2!\,3! = 1728$ .

(c) Only math together: $4!\,6! = 17280$ .

Seat $5$ people on a bench:

(a) No restriction: $5! = 120$ .

(b) Two particular people together: $4!\,2! = 48$ .

(c) Those two not together: $120 - 48 = 72$ .