AP DSC School Assistant Math Model Test 1

Concept:The number of ways to select items without regard to order is given by the combination formula: $nC_{r} = \frac{n!}{(n-r)! \, r!}$ This problem involves selecting 4 letters from the word “MATHEMATICS”, accounting for repeated letters.Explanation:The word “MATHEMATICS” has 11 letters: M (2), A (2), T (2), H (1), E (1), I (1), C (1), S (1). Thus there are 8 distinct letters, with 3 letters appearing twice. We consider three cases:Case 1 – All 4 letters different:Select 4 distinct letters from the 8 unique ones: $N_1 = {}^8C_4 = \frac{8!}{4!4!} = 70$Case 2 – Two pairs of identical letters (both pairs same):Choose 2 pairs from the 3 pairs of repeating letters: $N_2 = {}^3C_2 = 3$Case 3 – One pair of identical letters and two distinct different letters:Choose 1 pair from the 3 pairs: ${}^3C_1 = 3$ ways. Then choose 2 distinct letters from the remaining 7 distinct letters (since the letter used in the pair cannot be chosen again, and the other 7 are distinct): ${}^7C_2 = 21$ ways. So $N_3 = 3 \times 21 = 63$Total selections:$N_1 + N_2 + N_3 = 70 + 3 + 63 = 136$Answer:The total number of selections of four alphabets from “MATHEMATICS” is 136 (option D).