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

Question

The number of ways in which four letters of the word MATHEMATICS can be arranged, is given by (A) (8!/4! 4!) (B) 2454 (C) 2464 (D) 2474

Tardigrade Admin · Accepted Answer

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.

∴

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!

∴

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

.

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