Question

Find the number of all possible arrangement of the letters of the word "MATHEMATICS" taken four at a time.

Answer 1

We have 11 letters in "Mathematics"

in which there are

2 M's , 2 T's , 2 A's and other letters H,E,I,C,S are single

Selection of the 4 letters

first case: Two alike and other two alike

In this case we are gonna select the two letters which are alike. We have three choices M,T,A. Out of these, we have to select two(Because we have to select four letters and selecting two alike letters means selecting four letters). So, it can be done in 3C2 ways

second case: Two alike, two different

1 alike letter(which will mean two letters) can be selected in 3C1 ways and other 2 different letters can be selected in 7C2 ways.(as there will be 7 different letters).

So, 3C1*7C2 ways

Third case: All are different

This can be done in 8C4 ways as there are 8 different letters(M,T,A,H,E,I,C,S)

Arrangement:

For the first case, there will be two alike letters. So, arrangement of these letters can be done in 4!/2!*2! ways

For the second case, there will be 1 alike letter and two different letters which can be arranged in 4!/2! ways

For the third case, all letters are different so it can be arranged in 4! ways

So, the final answer will be

(3C2*4!/2!*2!) + (3C1*7C2*4!/2!) + (8C4*4!)

= 18+ 756 + 1680 ways

= 2454 ways