JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Permutations and Combinations

MATHEMATICS has letters: M(2), A(2), T(2), H(1), E(1), I(1), C(1), S(1) — 8 distinct letters, 3 of which repeat twice.
Choosing 5 letters (not necessarily distinct), by cases on how many repeated-letter pairs are used fully:

• All 5 distinct: choose 5 of the 8 distinct letters: $\left(\genfrac{}{}{0ex}{}{8}{5}\right)=56$.
• Exactly one letter used twice, other 3 distinct: choose the doubled letter (3 ways) and 3 more distinct letters from remaining 7: $3\times \left(\genfrac{}{}{0ex}{}{7}{3}\right)=3\times 35=105$.
• Exactly two letters used twice, 1 more distinct: choose 2 of the 3 repeatable letters and 1 more from remaining 6: $\left(\genfrac{}{}{0ex}{}{3}{2}\right)\times \left(\genfrac{}{}{0ex}{}{6}{1}\right)=3\times 6=18$.

Total $=56+105+18=179$.