Permutation and combination-Boys and Girls

s91arvindh wrote:Find the number of ways 5 boys and 4 girls can be seated in a row.
2 girls cannot sit together

Take the task of seating the 9 children and break it into stages.

Stage 1: Arrange the 5 boys in a row (in 5 ADJACENT chairs)
There's a nice rule that says we can arrange n unique objects in a row in n! ways.
So, we can complete stage 1 in 5! ways (120 ways)

IMPORTANT: Now place a chair on either side of each boy to get:
_B_B_B_B_B_
These are the "girl chairs" Notice that this ENSURES that no two girls sit together.

Stage 2: Seat one of the girls
There are 6 chairs, so we can complete this stage in 6 ways.

Stage 3: Seat another girl
There are now 5 chairs remaining digits, so we can complete this stage in 5 ways.

Stage 4: Seat another girl
There are now 4 chairs remaining digits, so we can complete this stage in 4 ways.

Stage 5: Seat the last girl
There are now 3 chairs remaining digits, so we can complete this stage in 3 ways.

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus seat all 9 children) in (120)(6)(5)(4)(3) ways ([spoiler]= 43,200 ways[/spoiler])

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Note: the FCP can be used to solve the majority of counting questions on the GMAT. For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html
- https://www.beatthegmat.com/mouse-pellets-t274303.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/ps-counting-t273659.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/please-solve ... 71499.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/laniera-s-co ... 15764.html

Cheers,
Brent