Question

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

• A. $\frac{8!}{4! 4!}$
• B. $2454$
• C. $2464$
• D. $2474$

Answer 1

Answer: (B)

Available letters are MMAATTHEICS
There are $11$ letters in which $2 M's, 2 A's$ and $2 T’s$.
The following cases are possible:
Case I : We have four letters having two pair of similar letters. This can be done in $^{3}C_{2}$ ways.
$\therefore$ Number of permutations of such four letters is $\frac{4!}{2! \,2!}$
Case II : One similar pair and $2$ different letters. This can be done in $^{3}C_{1} \times\,{}^{7}C_{2}$ ways and then number of possible permutations $\frac{4!}{ 2!}$
Case III : All different letters are taken. This can be done in $^{8}C_{4}$ ways and then number of permutations $= 4!$
$\therefore$ The required number is
$^{3}C_{2} \times\,\frac{4!}{2!\, 2!} +\,{}^{3}C_{1} \times\,{}^{7}C_{2} \times\, \frac{4!}{2!} +\,{}^{8}C_{4} \times\, 4!$
$= 2454$.