My Math Forum Select 5 balls from 12 balls, where some balls are identical

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 January 6th, 2014, 08:15 AM #1 Member   Joined: Jul 2011 From: Europe Posts: 59 Thanks: 2 Select 5 balls from 12 balls, where some balls are identical Box of 12 balls: 3 blue, 4 red, 5 green. In how many ways can 5 balls be selected? order of selection does not matter and the balls of same color are not distinguishable. I was told that the correct answer to this problem is 792, and I just can't think of why. The answer is equal to ncr(12,5), which would work if all the balls were unique (distinguishable), but they are not - right? Thanks for help.
 January 6th, 2014, 08:55 AM #2 Member   Joined: Apr 2013 Posts: 65 Thanks: 0 Re: Select 5 balls from 12 balls, where some balls are ident Are you sure the answer is 792? The only thing that matters is the amount of blue, red and green balls we choose. Thus we can write the equation $x_{b}+x_{r}+x_{g}=5$, where $x_{b}$ represents the number of blue balls chosen and so on. We know that $x_{b}\leq 3,\,x_{r}\leq 4$ and $x_{g}\leq 5$ If we didn't have these conditions, the equation would exactly have $7\choose2$ solutions for $x_{b},\,x_{r},\,x_{g}\geq0$ However we should substract the cases that are unlikely. There are exactly 4 of them, which are: $x_{b}= 4,\,x_{r} = 1,\,x_{g} = 0$ $x_{b}= 4,\,x_{r} = 0,\,x_{g} = 1$ $x_{b}= 5,\,x_{r} = 0,\,x_{g} = 0$ $x_{b}= 0,\,x_{r} = 5,\,x_{g} = 0$ Thus the answer is ${7\choose2} - 4= 17$ 0,0,5 0,1,4 0,2,3 0,3,2 0,4,1 1,0,4 1,1,3 1,2,2 1,3,1 1,4,0 2,0,3 2,1,2 2,2,1 2,3,0 3,0,2 3,1,1 3,2,0
 January 6th, 2014, 09:30 AM #3 Member   Joined: Jul 2011 From: Europe Posts: 59 Thanks: 2 Re: Select 5 balls from 12 balls, where some balls are ident Thank you very much, I was almost sure the answer I was given is wrong.
 January 6th, 2014, 11:26 AM #4 Member   Joined: Dec 2013 Posts: 82 Thanks: 0 Re: Select 5 balls from 12 balls, where some balls are ident $\dfrac{12!}{7!5!}=792$

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