The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to:

Solution

Maths sir · 4 steps

Sort the letters into pairs and singles first, then split by how many pairs you pick.

1. List the letters. MATHEMATICS has 11 letters but only 8 different ones. M, A and T appear twice each. H, E, I, C and S appear once each.
2. Shape of a choice. A letter can be used at most twice, and only if it is M, A or T. So a selection of 5 letters can contain 0, 1 or 2 repeated pairs. Three pairs would already need 6 letters.

   Case	How to choose	Ways
   All 5 different	any 5 of the 8 letters: C(8, 5)	56
   1 pair + 3 different	pair from M, A, T: C(3, 1); then 3 of the other 7 letters: C(7, 3)	3 × 35 = 105
   2 pairs + 1 different	pairs from M, A, T: C(3, 2); then 1 of the other 6 letters: C(6, 1)	3 × 6 = 18

3. Why exclude letters. Once a letter is used as a pair, both its copies are gone, so the remaining letters must come from the other ones. That is why it is 7 letters in the second case and 6 in the third.
4. So. Total = 56 + 105 + 18 = 179, which is option (a).

The idea to keep. When a word has repeated letters, sort them into pairs and singles, then break the selection into cases by how many repeated letters you use. Always take the remaining letters from the letters not already chosen.

Answer from the NTA final answer key. Solution by PrepWiser.