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Permutations and Combinations question

2017 · 9 Apr · Shift 1 · Q26
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  5. /2017 · 9 Apr · Shift 1 · Q26

Permutations and Combinations question

2017 · 9 Apr · Shift 1 · Q26

JEE MainMathematicsPermutations and CombinationsMCQ+4 / −1
The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B1 and a particular girl G1 never sit adjacent to each other, is :
  1. A
    5 ×\times× 6!
  2. B
    6 ×\times× 6!
  3. C
    7!
  4. D
    5 ×\times× 7!
View written solutionFree

Correct answer: A

  1. Total people and circular arrangement

There are 5+3=85+3=85+3=8 people.

The total number of ways to seat 888 distinct people around a round table is

(8−1)!=7!(8-1)! = 7!(8−1)!=7!
  1. Count arrangements where B1B_1B1​ and G1G_1G1​ are adjacent

If B1B_1B1​ and G1G_1G1​ must sit together, treat them as one block.

Then we have:

  • 111 block (B1,G1)(B_1,G_1)(B1​,G1​)
  • remaining 666 other people

So total units =7=7=7.

The number of circular arrangements of these 777 units is

(7−1)!=6!(7-1)! = 6!(7−1)!=6!

Within the block, B1B_1B1​ and G1G_1G1​ can be arranged in

2!2!2!

ways.

Hence, number of arrangements with B1B_1B1​ and G1G_1G1​ adjacent is

2⋅6!2\cdot 6!2⋅6!
  1. Required arrangements: not adjacent

Therefore,

Required=7!−2⋅6!\text{Required} = 7! - 2\cdot 6!Required=7!−2⋅6!

Now,

7!=7⋅6!7! = 7\cdot 6!7!=7⋅6!

So,

7!−2⋅6!=(7−2)6!=5⋅6!7! - 2\cdot 6! = (7-2)6! = 5\cdot 6!7!−2⋅6!=(7−2)6!=5⋅6!
  1. Match with options

Thus the correct answer is

5×6!5\times 6!5×6!

which is Option A.

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