dabaobao wrote:

Seven women and four men have to sit around a circular table so that no two men are together. In how many different ways can this be done?

A) 17,280

B) 25,200

C) 103,680

D) 604,800

E) 3,628,800

Official Solution

Credit:

Veritas PrepTry and think about it for a while. We did a very similar question while working on circular arrangements. In that question, number of women and number of men were equal so we just had to place them in alternate positions. Here, we have fewer men. What do we do now?

Two men cannot sit together but some women will sit together since there aren’t enough men. So, let’s make the 7 women sit around the round table in (7-1)! = 6! ways (We covered the (n-1)! concept in the post on circular arrangements)

Now, how many places do we have for the men? A man can sit between any two women sitting next to each other. How many such pairs of women are there? Since there are 7 women, we have 7 such pairs and hence 7 possible spaces for men. There are two different approaches you can take from here:

Approach 1:

We have 4 men but 7 possible spaces for them. For the first man, we can select a space in 7 ways. For the second man, we can select a space in 6 ways. For the third one, in 5 ways and for the fourth one in 4 ways. So we can arrange the men in 7*6*5*4 ways. This is just our basic counting principle in action.

Approach 2:

Some people like to split up the task into two steps – make the selection, then arrange. Out of 7 spaces, we need to select any 4 for the 4 men. How do you select 4 out of 7? Using basic counting principle and un-arranging concept, we can do it in 7*6*5*4/4! ways (or we can use the formula 7C4). We have selected 4 spaces so now we just want to arrange the 4 men in the 4 spaces. We can do this in 4! ways.

It doesn’t matter which approach you use. The first one uses just the basic counting principle. The second one is used more often by people who are very comfortable with the combinations formula.

The total number of arrangements we get = 6! * 7*6*5*4 or we can write this as 6!*7!/3! to make it a little compact.

6! * 7*6*5*4 = 604,800

ANSWER=D

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