Students Standing in Circle!

Probability Level 3

4 girls and 8 boys are standing together. In how many ways can a 7- person committee be selected from the group if at least 2 girls must be included?


The answer is 672.

Samara Simha Reddy by Samara Simha Reddy , 22, India

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1 solution

Relevant wiki: Combinations without Repetition - Basic

We can choose 2,3 or 4 girls and 5,4 or 3 boys. Number of ways = ( 4 2 ) × ( 8 5 ) + ( 4 3 ) × ( 8 4 ) + ( 4 4 ) × ( 8 3 ) = 4 ! 2 ! × 2 ! × 8 ! 3 ! × 5 ! + 4 ! 3 ! × 8 ! 4 ! × 4 ! + 4 ! 4 ! × 8 ! 3 ! × 5 ! = 336 + 280 + 56 = 672 . \large \displaystyle \text{We can choose 2,3 or 4 girls and 5,4 or 3 boys.}\\ \large \displaystyle \text{Number of ways } = \dbinom{4}{2} \times \dbinom{8}{5} + \dbinom{4}{3} \times \dbinom{8}{4} + \dbinom{4}{4} \times \dbinom{8}{3}\\ \large \displaystyle = \frac{4!}{2! \times 2!} \times \frac{8!}{3! \times 5!} + \frac{4!}{3!} \times \frac{8!}{4! \times 4!} + \frac{4!}{4!} \times \frac{8!}{3! \times 5!}\\ \large \displaystyle = 336 + 280 + 56 = \color{#3D99F6}{\boxed{672}}.

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Nice Solution.. +1

Sabhrant Sachan - 5 years, 1 month ago

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Thanks! ⌣ ¨ \Large \ddot\smile

Samara Simha Reddy - 5 years, 1 month ago

Solved it the same way. :) (･ิω･ิ) (+1)

Ashish Menon - 5 years ago

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