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Oct 29, 2019
edited by sinclairdragon428  Nov 20, 2019

#1
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The Smith family has 4 sons and 3 daughters. In how many ways can they be seated in a row of 7 chairs such that all 3 girls sit next to each other?

Number the chairs 1, 2, 3, 4, 5, 6, and 7

The girls will be seated together if they're in chairs  1, 2, 3

2, 3, 4

3, 4, 5

4, 5, 6

or     5, 6, 7

unless you arrange the chairs in a circle

and then you can add to the above                          6, 7, 1

7, 1, 2

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Oct 29, 2019
#2
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I wasn't through.  I was previewing the text and then clumsily clicked Publish instead of Edit.

To finish:

There are 5 seating positions, in the straight row.

The girls can be arranged in 6 ways in each position.

So there would be 30 ways the girls could be seated.

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Guest Oct 29, 2019
#3
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There are 7! ways to arrange all 7 children, but there are 3! ways that the girls can sit next to each other, so the number of arrangement is 7! - 3! = 5034.

Oct 29, 2019