Question

How many permutations of $4$ letters can be made out of the letters of the word examination?

Answer 1

$⟹$ $\underset{–}{e}$ $\underset{–}{x}$ $\underset{–}{aa}$ $\underset{–}{m}$ $\underset{–}{ii}$ $\underset{–}{nn}$ $\underset{–}{t}$ $\underset{–}{o}$

Total number of single letters $=8$ $(e,x,a,m,i,n,t,o)$
Duplicate letters $=3(aa,ii,nn)$
Possible ways of selecting $4$ letters are :-
$4$ from single ${=}^8{C}_4\times 4!=1680$
$2$ from single and $1$ pair of the duplicated letters $={}^3{C}_1\times {}^7{C}_2\times \frac{4!}{2!}=756$
$2$ pair of duplicated letters $={}^3{C}_2\times \frac{4!}{2!2!}=18$
Total no. of ways $=2454$