b- are you sure that 144 is correct?

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Posted on 2011-10-28

The school board consists of three men and four women

a. When they hold a meeting, they sit in a row. How many different seating arrangements are there ? Ans. 5,040

b. How many different ways can the row be arranged if no two women sit next to each other? Ans. 144

Can any one please explain how to get these numbers ? Thanks.

a. When they hold a meeting, they sit in a row. How many different seating arrangements are there ? Ans. 5,040

b. How many different ways can the row be arranged if no two women sit next to each other? Ans. 144

Can any one please explain how to get these numbers ? Thanks.

9 Comments

b- are you sure that 144 is correct?

first, lets look at just Male/Female arrangements, there are only 10 of those that are legal.

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so, now it's simply a matter of how many ways can you arrange the men and women within each of those 10, THAT is 144.

4! * 3! (24 ways to arrange 4 men and 6 ways to arrange 3 women)

so, 144 * 10 = 1440 total possible arrangements

for 3men, 4 women there is only one arrangement

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so, 4! x 3! x 1 = 144

sorry for the confusion

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but in the question it says 3 man and 4 woman

How many different ways can the row be arranged if no two women sit next to each other?

for 3men, 4 women there is only one arrangement

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so, 4! x 3! x 1 = 144

1 - there is only one arrangement, that should hopefully be obvious. I used no math for that, I simply listed possibilities until I ran out, and I ran out of options after 1.

4! - this is the number of ways the women can be arranged in the row (we're ignoring the men here) - this is the same logic as the 7! in part A

3! - this is the number of ways the men can be arranged in the row (we're ignoring the women here) - again, same logic as the 7! in part A

we multiply them together because for each arrangement of women you can have each arrangement of men - their individual seating is not dependent on the other.

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