JEE Maths›Permutations & Combinations›Visual Solution
Visual SolutionPYQ 2024 · Jan 31 Shift 2Easy
Question
The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
(A)406406
(B)130130
(C)142142
(D)136136
Solution Path
Give 2 each first →\to 15 remaining →\to stars and bars 17C2=136{}^{17}C_2 = 136
01Read the Problem
1/4
21 identical apples, 3 children, each must get at least 2. How many distributions?
Distribute with minimum constraint
02Give Minimum First
2/4
Give 2 apples to each child upfront. Used: 3×2=63 \times 2 = 6. Now distribute the remaining freely.
21−6=1521 - 6 = 15 apples remaining
03Stars and BarsKEY INSIGHT
3/4
15 identical objects into 3 groups with no restriction. Apply the stars and bars formula: n+r−1Cr−1{}^{n+r-1}C_{r-1}.
17C2=17×162=136{}^{17}C_2 = \dfrac{17 \times 16}{2} = 136
04Final Answer
4/4
Give minimum first, then stars and bars on the remainder.
136\boxed{136} ways
Concepts from this question3 concepts unlocked
★

Stars and Bars

EASY

Distribute n identical objects into r distinct groups: C(n+r-1, r-1)

n+r−1Cr−1=n+r−1Cn{}^{n+r-1}C_{r-1} = {}^{n+r-1}C_{n}

The go-to formula for identical object distribution. Appears in 2-3 JEE questions every year.

Distribution problemsInteger solutionsPartition counting
Practice (10 Qs) →
★

Minimum Constraint Reduction

EASY

If each group needs at least k objects, give k to each first, then distribute the remainder freely

xi≥k  ⟹  yi=xi−k≥0,new total=n−rkx_i \geq k \implies y_i = x_i - k \geq 0,\quad \text{new total} = n - rk

Converts a constrained problem into an unconstrained one. Without this trick, students try inclusion-exclusion and waste time.

At-least constraintsDistribution with lower boundsInteger equation solutions
Practice (5 Qs) →

Identical vs Distinct Objects

EASY

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

Identical: n+r−1Cr−1Distinct: rn or nPr\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
Practice (7 Qs) →
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