Permutations & CombinationsmediumPYQ · JEE Main · 5 Apr 2026 · Shift 2 (Afternoon)Free

Permutations & Combinations: Box Contains Blue Yellow Red Balls Number Ways (JEE Main 2026)

JEE Maths question with a full step-by-step solution.

Question
A box contains 5 blue, 6 yellow and 4 red balls. The number of ways of drawing 8 balls containing at least two balls of each colour is:
A41004100correct
B41404140
C42304230
D42904290
Solution
Step 1: Enumerate valid distributions With b+y+r=8b+y+r=8 and b≥2b\geq2, y≥2y\geq2, r≥2r\geq2 subject to b≤5b\leq5, y≤6y\leq6, r≤4r\leq4:
{(b,y,r)}={(2,2,4), (2,3,3), (2,4,2), (3,2,3), (3,3,2), (4,2,2)}\{(b,y,r)\} = \{(2,2,4),\ (2,3,3),\ (2,4,2),\ (3,2,3),\ (3,3,2),\ (4,2,2)\}
 Step 2: Count the combinations for each distribution
(52)(62)(44)=150,(52)(63)(43)=800,(52)(64)(42)=900\binom{5}{2}\binom{6}{2}\binom{4}{4} = 150, \quad \binom{5}{2}\binom{6}{3}\binom{4}{3} = 800, \quad \binom{5}{2}\binom{6}{4}\binom{4}{2} = 900
(53)(62)(43)=600,(53)(63)(42)=1200,(54)(62)(42)=450\binom{5}{3}\binom{6}{2}\binom{4}{3} = 600, \quad \binom{5}{3}\binom{6}{3}\binom{4}{2} = 1200, \quad \binom{5}{4}\binom{6}{2}\binom{4}{2} = 450
 Step 3: Sum all cases
150+800+900+600+1200+450=4100150+800+900+600+1200+450 = 4100
 Answer: (1)
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