Question

In how many different ways can the letters of the word $MATHEMATICS$ be arranged?

• A. $11!$
• B. $\frac{11!}{2!}$
• C. $\frac{11!}{{(2!)}^2}$
• D. $\frac{11!}{{(2!)}^3}$

Answer 1

The correct option is D

$\frac{11!}{{(2!)}^3}$

Explanation of the correct option :

First, find the number of ways in which the word $‘Mathematics’$ can be written, and then we use the permutation formula with repetition which is given as
Number of permutation of $n$objects with$n_1$ identical objects of type,$1,n_2$identical objects of type $2$…….,$n_k$identical objects of type $k$ is

$\frac{\mathbf{n}\mathbf{!}}{{\mathbf{n}}_{\mathbf{1}}\mathbf{!}{\mathbf{n}}_{\mathbf{2}}\mathbf{!}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{n}}_{\mathbf{k}}\mathbf{!}}$

Total number of letters in $MATHEMATICS=11$

There are $2A’s,1C,1E,1H,1I,2M’s,1S,2T$

Total number of different words $=\frac{11!}{2!\times 2!\times 2!}=\frac{11!}{{(2!)}^3}$

Hence, the correct option is D.