Question

The number of words of four letters that can be formed from the letters of the word EXAMINATION is

• A. $1464$
• B. $2454$
• C. $1678$
• D. None of these

Answer 1

The correct option is B $2454$

There are 11 letters A, A; I, I;N, N;E, X, M, T, O. For the selection of 4 letters we have the following possibilities:
(A) 2 alike, 2 alike
(B) 2 alike, 2 different
(C) All four different

(A) There are 3 pairs of 2 letters. So, the number of ways of selection of 2
pairs is ${}^3{C}_2$ and permutation of these 4 letters is
$\frac{4!}{2!2!}$ Therefore, the number of words in
this case is ${}^3{C}_2\times \frac{4!}{2!2!}=18$.
(B) We have to select one pair from 3 pairs and 2 distinct letters from
remaining 7 distinct letters. For illustration, let us select both A, A;
then we have I, N, E, X, M, T, O, i.e., 7 as remaining distinct
letters. Hence, the number of selections is ${}^3{C}_1{\times }^7{C}_2$ and
these 4 (2 same, 2 distinct) can be permuted in
$\frac{4!}{2!}$ ways. Therefore, number of words is
${}^3{C}_1{\times }^7{C}_2\times \frac{4!}{2!}=3\times 21\times 12=756$.
(C) There are 8 distinct letters so number of words of 4
letters is ${}^8{C}_4\times 4!=1680$.
$\therefore$ By sum rule, the total number of
words is $18+756+1680=2454$.