How many ways we can select four letters from the word mathematics?

We have 11 letters in "Mathematics"

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 = 3 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 = 63 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) = 70 ways.