JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Permutations and Combinations

The letters of DISTRIBUTION are D, I, S, T, R, I, B, U, T, I, O, N — 12 letters with I repeated 3 times, T repeated 2 times, and 7 other distinct letters (D, S, R, B, U, O, N), giving 9 distinct letter-types in total.

3 alike + 1 different (only I has 3 copies): choose the other letter from the remaining 8 types, arrange in $\frac{4!}{3!}=4$ ways: $8\times 4=32$

2 alike + 2 alike (only I and T have repeats): only 1 way to choose both pairs (I,I and T,T), arrange in $\frac{4!}{2!2!}=6$ ways: $1\times 6=6$

2 alike + 2 different: choose which letter repeats (I or T, 2 ways), choose 2 different letters from the remaining 8 types ($\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$), arrange in $\frac{4!}{2!}=12$ ways: $2\times 28\times 12=672$

All 4 different: choose 4 distinct types from the 9 available ($\left(\genfrac{}{}{0ex}{}{9}{4}\right)=126$), arrange in $4!=24$ ways: $126\times 24=3024$

Total: $32+6+672+3024=3734$