Question:

The word Calculus has 8 letters: C, A, L, C, U, L, U, S (with C appearing twice, L appearing twice, U appearing twice). We form 5-letter words by selecting 5 letters from these 8. The total count involves cases: (i) All different: $\binom{5}{5} \times 5! = 120$; (ii) One letter twice, three different: $\binom{3}{1} \times \binom{4}{3} \times \frac{5!}{2!} = 3 \times 4 \times 60 = 720$; (iii) Two letters twice each, one different: $\binom{3}{2} \times \binom{3}{1} \times \frac{5!}{2! \times 2!} = 3 \times 3 \times 30 = 270$. Total = $120 + 720 + 270 = 1110 = 2 \times 3 \times 5 \times 37$. Thus 1110 is divisible by 2, 3, and 5.