Question

The number of permutations of 4 letters that can be made out of the letters of the word EXAMINATION is

• A. $ 2454 $
• B. $ 2452 $
• C. $ 2450 $
• D. $ 1806 $

Answer 1

The Correct Option is A

In a word EXAMINATION has $ 2A,\,2I,\,2N,\,E,\,M,\,O,\,T,\,X, $ therfore 7 letters can be chosen in following ways Case I when 2 alike of one kind and 2 alike of second kind of is
$ ^{3}{{C}_{2}} $ .
$ \therefore $ Number of words $ {{=}^{3}}{{C}_{2}}\times \frac{4!}{2!\,\,2!}=18 $
Case II when 2 alike of one kind and 2 different ie,
$ ^{3}{{C}_{1}}{{\times }^{7}}{{C}_{2}}. $
$ \therefore $ Number of words $ {{=}^{3}}{{C}_{1}}{{\times }^{7}}{{C}_{2}}\times \frac{4!}{2!} $
$ =756 $ Case III when all are different. ie, $ ^{8}{{C}_{4}} $ .
$ \therefore $ Number of words $ {{=}^{8}}{{C}_{4}}\times 4!=1680 $
Hence, total number of words $ =18+756+1680 $
$ =2454 $