Problem Solving

can some one please explain this question, i cant understand why 6! for the men??

also please suggest some good resource for combination permutation questions i am having diffculty solving them.

thanks



Seven men and seven women have to sit around a circular table so that no 2 women are together. In how many different ways can this be done?
a.24 b.6 c.4 d.12 e.3

Soln: I suggest to first arranging men. This can be done in 6! Ways. Now to satisfy above condition for women, they should sit in spaces between each man. This can be done in 7! Ways (because there will be seven spaces between each man on round table)
Total ways = 6! * 7!

This is the question of circular permutations

therefore the formula is (n-1)!

For example,
if you want to arrange 5 people in a circular table, how many ways can you do that?

Since we are dealing with circular position, you fix 1 person, which can be done in 1 way. Now arrange 4 people in the circle. This can be done in 4! ways

total no. of ways = 4! * 1 = 4! = 24, which is same as the formula, (5-1)! = 4! = 24


Now coming to the question:

arrange 7 men and 7 women, so that no 2 women are sitting together.

This can be done in 2 ways, logic is the same though

Method 1

Fix 1 man, remaining 6 men can be arranged in 6! ways

Now arrange 7 women between 6 men, this can be done in 7! ways

total ways = 6! * 7!

Method 2

Fix 1 woman, remaining 6 women can be arranged in 6! ways
now arrange 7 men between 6 women so that no 2 women can sit together = 7! ways

total ways = 7! * 6!

Hope this helps.

thanks a lot ,

I understand the explanation, but why doesn't it match up with any of the answer choices?

jsk988 wrote:I understand the explanation, but why doesn't it match up with any of the answer choices?

I believe the answer options are incorrect.