(Q). In a class of 12 students 7 are boys and rest are girls. In how many ways students can be selected for a project such that at least 2 boys & 2 girls are selected?
Solution:-
Number of ways of selecting at least 2 boys
= ⁷C₂ + ⁷C₃ + ⁷C₄ + ⁷C₅ + ⁷C₆ + ⁷C₇
= (⁷C₀ + ⁷C₁ + ⁷C₂ + ⁷C₃ + ⁷C₄ + ⁷C₅ + ⁷C₆ + ⁷C₇) – ⁷C₀ – ⁷C₁
= 2⁷ – 1 – 7
= 120
Similarly, number of ways of selecting at least 2 girls = ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅
= (⁵C₀ + ⁵C₁ + ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅) – (⁵C₀ + ⁵C₁)
= 2⁵ – 1 – 5
= 26
Hence total number of ways = 120×26
= 3120 Answer