Question

Number of words that can be formed by taking $4$ letters at a time out of the letters of the word $MATHEMATICS$ is

• A. $1680$
• B. $1698$
• C. $2436$
• D. $2454$

Answer 1

The correct option is D $2454$

There are $8$ distinct letters: $M,A,T,H,E,I,C,S$. $3$ letters $M$, $A$, and $T$ are represented twice (double letter).

Selected $4$ letters can have following $3$ patterns:

Case $1:$ all $4$ letters are different:
${}^8{P}_4=1680$
Case $2:$ from $4$ letters $2$ are the same and other $2$ are also the same:
${}^3{C}_2\times \frac{4!}{2!2!}=18$

Case $3:$ from $4$ letters $2$ are the same and other $2$ are different:
$\left({}^3{C}_1{\times }^7{C}_2\right)\frac{4!}{2!}=756$

Now total number of ways:
$1680+18+756=2454$