Question

A committee of 4 students is selected at random from a group of consisting of 7boys and 4 girls.find the probability that there are 2 boys exactly in the committee then atleast one girl must be there in the committee

Answer 1

(a) when committee of 4 student is selected exactly two boys , it means rest two selection only for girl .
So, number of way of selection of boys = ⁷C₂
number of way of girl of selection of girls = ⁴C₂
So, favourable outcomes = ⁷C₂ × ⁴C₂
Sample space = ¹¹C₄

Hence, P(E) = ⁷C₂ × ⁴C₂/¹¹C₄ = (7!/5!×2!)(4!/2!2!)/(11!/7!×4!)
= (7 × 6 × 3 )/(11 × 10 × 3) = 126/330 = 21/55

(B) at least one girl is selected in committee
There are four cases possible ,
case 1 :- committee selects 1 girl , 3 boys
number of ways = ⁴C₁ × ⁷C₃ = 35/6

Case 2 :- committee selects 2 girls , 2 boys
Number of ways = ⁴C₂ × ⁷C₂ = 6 × 21 = 126

case 3 :- committee selects , 3 girls , 1 boy
Number of ways = ⁴C₃ × ⁷C₁ = 4 × 7 = 28

Case 4 :- committee selects , 4 girls , 0 boy
number of ways = ⁴C₄ × ⁷C₀ = 1

And sample space = ¹¹C₄ = 330

Now, P(E) = (35/6 + 126 + 28 + 1)/330

Answer 2

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Please see the attached file !!

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