JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Permutations and Combinations

Let

n

be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let

m

be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of

\frac{m}{n}

Answer: 5

Solution

Case 1 (all 5 girls consecutive): Treat the 5 girls as one unit. We have 5 boys + 1 unit = 6 objects to arrange in

6!

ways. The 5 girls within their unit can be arranged in

5!

ways.

n = 6! \times 5!

Case 2 (exactly 4 girls consecutive): Choose 4 girls from 5:

(4 5) = 5

ways. Treat these 4 as one unit. We have 5 boys + 1 unit + 1 remaining girl = 7 objects. The unit and the single girl must NOT be adjacent. Arrange 5 boys in

5!

ways, creating 6 gaps. Choose 2 non-adjacent positions from these 6 gaps:

(2 6) = 15

ways. The 4 girls in the unit can be arranged in

4!

ways, and we can assign which position gets the unit vs. the single girl in

2!

ways.

m = 5 \times 5! \times 15 \times 4! \times 2!

\frac{m}{n} = \frac{5 \times 5 ! \times 15 \times 4 ! \times 2}{6 ! \times 5 !} = \frac{5 \times 15 \times 2}{6} = \frac{150}{30} = 5

JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Integer answer

Practise more: all JEE Advanced PYQs on Permutations and Combinations →

JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Permutations and Combinations

Solution

More JEE Advanced questions on Permutations and Combinations