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16 people to choose from

by sanju09 » Wed Sep 15, 2010 9:36 pm
If there are 16 people to choose from, what is the ratio of the number of possible 7-person committees to the number of possible 8-person committees?
(A) 7:8
(B) 8:7
(C) 7:9
(D) 8:9
(E) 9:8



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by Rahul@gurome » Wed Sep 15, 2010 11:14 pm
7-person committee can be formed in 16C7:16C8 = 16!/(7!)(9!) : 16!/(8!)(8!) = 8/9

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by goyalsau » Fri Sep 17, 2010 12:18 am
Thanks Rahul to such a short cut i was thinking what could be the best way to solve it.
Nice work you made lengthy calculation very easy..
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by alindawit » Sun May 29, 2011 12:22 am
Hi,

Thanks for the answer, but could you explain how you can quickly get to 8/9? From 16!/7!9! / 16!/8!8! I do not immediately retrieve 8/9....

Thanks!
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by SoCan » Sun May 29, 2011 9:37 am
alindawit wrote:Hi,

Thanks for the answer, but could you explain how you can quickly get to 8/9? From 16!/7!9! / 16!/8!8! I do not immediately retrieve 8/9....

Thanks!
I think it's easiest to see when you put the ratio in fraction form and write it out. Hopefully the formatting works out here:

If you put it as a fraction and flip the bottom term to multiply, you get
16! 8!8!
---- * ----
7!9! 16!

Now you can see that the 16!s cancel, the 7! cancels out all terms of 8! except 8, and the other 8! cancels out all the terms of 9! except 9.

EDIT: The formatting doesn't work, but hopefully you get the picture.
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