JEE Maths›Permutations & Combinations›Visual Solution

Visual SolutionPYQ 2024 · Jan (1 Feb) Shift 1Standard

Question

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:

(A)

47

(B)

53

(C)

51

(D)

43

Solution Path

Set partition of 5 into at most 4 groups:

1+5+10+10+15+10 = 51

01Question Setup

1/4

Assign 5 employees to 4 indistinguishable offices (each office has at least one employee is not required). Find the number of ways.

\text{Ways} = \;?

02Set Partitions

2/4

Indistinguishable offices means set partitions. Partition 5 people into at most 4 non-empty groups.

S(5, k)

for

k = 1, 2, 3, 4

03CountingKEY INSIGHT

3/4

Partition types:

\{5\}

: 1,

\{4,1\}

: 5,

\{3,2\}

: 10,

\{3,1,1\}

: 10,

\{2,2,1\}

: 15,

\{2,1,1,1\}

: 10. Total

= 51

1 + 5 + 10 + 10 + 15 + 10 = 51

04Final Answer

4/4

All set partitions of 5 into at most 4 groups sum to 51.

\boxed{51}

Concepts from this question1 concepts unlocked

★

Identical vs Distinct Objects

EASY

Identical objects use combinations (stars and bars). Distinct objects use permutations or multiplication principle.

\text{Identical: } {}^{n+r-1}C_{r-1} \quad \text{Distinct: } r^n \text{ or } {}^nP_r

The first decision in every distribution problem. Wrong classification = completely wrong approach.

DistributionArrangementSelection problems

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