Question

The number of ways in which four letters of the word MATHEMATICS can be arranged is given by:

• A. $136$
• B. $192$
• C. $1680$
• D. $2454$

Answer 1

Answer: (D)

$2454$

Two pairs of identical letters can be arranged in ${}^3{C}_2\frac{4!}{2!2!}$ ways.

Two identical letters and two different letter can be arranged in ${}^3{C}_1\times {}^7{C}_2\times \frac{4!}{2!}$ ways.

All different letters can be arranged in ${}^8{P}_4$ ways.
$\therefore$ Total number of arrangements
$={}^3{C}_2\frac{4!}{2!2!}+{}^3{C}_1\times {}^7{C}_2\times \frac{4!}{2!}+\frac{8!}{4!}=2454$.