Question

A set has $5$ elements. Let $n$ be the number of ways of partitioning the set into $2$ or more non-empty subsets. Then $n$ is

• A. divisible by $3$
• B. divisible by $17$
• C. divisible by $13$
• D. a prime number

Answer 1

Answer: (A), (B)

The correct options are
A divisible by $3$
B divisible by $17$

Partition means partition into disjoint subsets.
Case-1: Partition into 2 subsets
1 element 4 elements
or 2 elements 3 elements
Number of ways of forming subsets ${=}^5{C}_4{+}^5{C}_3=15$
Case 2: Partition of 3 subsets
1 1 3
1 2 2

Number of ways of forming subsets ${=}^5{C}_3{.}^2{C}_1{+}^5{C}_2{.}^3{C}_2=50$
Both ways include double counting of subsets.
Number of ways of forming subsets $=\frac{50}{2}=25$
Case 3: Partition into 4 subsets
1 1 1 2
Number of ways of forming subsets ${=}^5{C}_2=10$
Case 4: Partition of 5 subsets (one element in each) $=1$
Hence, $n=15+25+10+1=51$
It is divisible by $3$ and $17$.