Question

How many different three letter arrangements can you make from the word: MATHEMATICS?

Answer 1

MATHEMATICS has 11 letters of which there are 2 M's, 2 A's and 2 T's.

There are 8 distinct alphabets (MATHEICS) of which 3 alphabets M, A and T get repeated and 5 are appearing once.

The 3 letter word can be formed in 2 ways:

(i) All 3 letters are distinct: We choose 3 alphabets out of out of 8 in ⁸C₃ ways and arrange them in 3! ways

= ⁸C₃ * 3! = 56 * 6 = 336 words

(ii) 2 letters are same and the third is different: Choose 1 alphabet out of the 3 repeating alphabets in ³C₁ ways. Thi8s takes care of 2 letters. The 3rd letter can be chosen from the 7 distinct letters in ⁷C₁ ways. They are then arranged in 3!/2! ways (as 2 of the letters are the same. Therefore total words in this scenario = ³C₁ * ⁷C₁ * (31/2!) = 3 * 7 * 3 = 63 words

Therefore the total number of words = 336 + 63 = 399 words