Permutations & Combinations
Combinations with multiple constraints
nta_pyq_2023_jan
Grade 11
Question:
Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways Anil's mother can offer 5 fruits to Anil is
Step-by-Step Solution
Key Concept: Use casework: exactly 2 oranges or 3 oranges with required minimums of red and white apples.
= ⁸C₂$\times$⁷C₁$\times$⁵C₂ + ⁸C₂$\times$⁷C₂$\times$⁵C₁ + ⁸C₃$\times$⁷C₁$\times$⁵C₁ = 28$\times$7$\times$10 + 28$\times$21$\times$5 + 56$\times$7$\times$5 = 1960 + 2940 + 1960 = 6860.
Correct Answer: 6860