JEE Main 2024 (Online) 1st February Morning ShiftQuestion 1
If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to :
Permutation and Combination
Solution
Concept. The offices are identical, so only the grouping of the 5 people matters. Empty offices are allowed, so we may use 1, 2, 3 or 4 groups. Formulas used. (1) $S(n,k)$ = ways to split $n$ distinct people into $k$ non-empty unlabelled groups(2) Number of ways $=S(5,1)+S(5,2)+S(5,3)+S(5,4)$Step 1 - $S(5,1)$. Everyone together: $1$. Step 2 - $S(5,2)$. Sizes $4+1$: $5$; sizes $3+2$: $\binom52=10$. Total $15$. Step 3 - $S(5,3)$. Sizes $3+1+1$: $\binom53=10$; sizes $2+2+1$: $\frac{5\cdot\binom42}{2}=15$. Total $25$. Step 4 - $S(5,4)$. Sizes $2+1+1+1$: $\binom52=10$. Step 5 - Add. $1+15+25+10=51$. Answer: A