Permutations & Combinations
Choosing Letters from MATHEMATICS — With Repetition Allowed
nta_pyq_2024_apr
Grade 11
Question:
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:
Step-by-Step Solution
Key Concept: MATHEMATICS has letters: M(×2), A(×2), T(×2), H, E, I, C, S — 8 distinct types, 3 pairs. Cases: (1) All distinct: $^8C_5=56$. (2) Exactly one pair + 3 different: $^3C_1\times^7C_3=3\times35=105$. (3) Two pairs + 1 different: $^3C_2\times^6C_1=3\times6=18$.
$^8C_5+^3C_1\cdot^7C_3+^3C_2\cdot^6C_1=56+105+18=179$.
Correct Answer: 1