Question

In how many ways we can select $3$ letters of the word $PROPORTION$?

Answer 1

In proportion

$PP,RR,OOO,T,I,N$

Total no. of ways all $3$ are different $=6_{C_3}$

Total no. of ways $2$ alike and $1$ distinct $=3_{C_1}\times 5_{C_1}$

Total no. of ways all $3$ are alike i.e; $=1[\because OOO]$

$\therefore$ Total ways ${=}^6{C}_3{+}^3{C}_1{\times }^5{C}_1$

$=\frac{6!}{3!\times 3!}+3\times 5+1$

$=\frac{4\times 5\times 6}{3\times 2}+15+1$

$=20+15+1=36$ ways.