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Old 3rd June 2012, 07:44 PM   #1 (permalink)
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P&C Question

a club has 11 members: 6 male and 5 female. 4 members are to be chosen for a competition. how many ways can this be done if there must be at least one male and one female?

my issue here is not getting the right answer as i do have the solution and i understand the method presented by the solution, which is:

number of ways = (total number of ways to pick 4 members without restrictions) - (number of ways without males) - (number of ways without females)

the way i attempted to solve the question initially though was to take 6C1 x 5C1 x 9C2 = 1080. obviously the answer is wrong since it's more than the number of ways without restriction which is 11C4 = 330. but what i do not understand is what is fundamentally wrong with my method. it seems correct -- i pick one male out of the 6 males, one female out of the 5 females (thus fulfilling the requirements) and then proceed to pick 2 other members out of the remaining 9.

any explanations? i have a feeling this is an issue with my core understanding of how p&c works but i can't seem to identify what.

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Old 3rd June 2012, 08:04 PM   #2 (permalink)
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Re: P&C Question

Let's name the males A, B, C, D, E, F
and the females V, W, X, Y, Z.

Let's use your method to take a look at the selection.

COMBINATION 1:
First pick one from the males. Say, we pick A.
Then pick one from the females. Say, we pick V.
Then we pick two more from the remaining. Say, we pick C and X.
So our combination 1 is A, V, C, and X.

Next combination, COMBINATION 2:
First pick one from the males. Say, we pick C.
Then pick one from the females. Say, we pick X.
Then we pick two more from the remaining. Say, we pick A and V.
So our combination 2 is C, X, A and V.

Eh? You picked the same 4!
Although its different way of picking, you end up with the same combination.
You have to deduct away so many of the "same selections" from your method, say, D, W, X, Y and D, Y, X, W, and we have no idea how many are there.
A better way would of course be, taking the (total) - (restrictions).

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Old 3rd June 2012, 08:27 PM   #3 (permalink)
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Re: P&C Question

ahh, can't believe i forgot about the possibility of repeat combinations. thanks a lot, that was a really clear and great explanation. much appreciated!

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Old 3rd June 2012, 10:55 PM   #4 (permalink)
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Re: P&C Question

Originally Posted by zette View Post
ahh, can't believe i forgot about the possibility of repeat combinations. thanks a lot, that was a really clear and great explanation. much appreciated!
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