JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Permutations and Combinations

Distributing 5 distinct people into 4 identical (indistinguishable) offices, offices allowed to be empty, is equivalent to summing over every way of partitioning 5 into at most 4 positive parts (the group sizes) and distributing distinct people into groups of those sizes:

• $5$: $\left(\genfrac{}{}{0ex}{}{5}{5}\right)=1$ way
• $4,1$: $\left(\genfrac{}{}{0ex}{}{5}{4}\right)=5$ ways
• $3,2$: $\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$ ways
• $3,1,1$: $\frac{5!}{3!1!1!2!}=10$ ways
• $2,2,1$: $\frac{5!}{2!2!1!2!}=15$ ways
• $2,1,1,1$: $\frac{5!}{2!1!1!1!3!}=10$ ways

Total: $1+5+10+10+15+10=51$