Question

A five letter word is to be formed such that the letters appearing in the word $MATHEMATICS$.

Further, the letters appearing in the even numbered positions taken from the letters which appear with repetitions in the same word $MATHEMATICS$.

The number of different ways in which the five letter word be formed is

• A. $390$
• B. $600$
• C. $360$
• D. $450$

Answer 1

The correct option is C

$360$

Explanation for the correct option:

Step1. Finding the number of ways

Given word is $MATHEMATICS$

Words without repetition $=H,E,\hspace{0.17em}C,I,S$

Repetition Letter$=M,A,\hspace{0.17em}T$

For five letter words, we have three odd positions and two even position.

As $5$ words is non repeated and out of which $3$ is taken at a time that is ${}^{5}C_3$ and to fill three odd places number of ways they arrange themselves is $3!$

Number of ways to fill odd positions without repeated words is ${}^{5}C_3\times 3!=60$

As $3$ words is repeated and out of which $2$ is taken at a time that is ${}^{3}C_2$ and to fill two even places number of ways they arrange themselves is $2!$

Number of ways to fill even positions with repeated words is ${}^{3}C_2\times 2!=6$

Therefore, the total number of ways $=60\times 6$

$\mathbf{=}\mathbf{360}$

Hence, the correct option is $\mathbf{\left(}\mathit{C}\mathbf{\right)}$