My Math Forum Combination containing a pair

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 January 13th, 2016, 04:19 AM #1 Newbie   Joined: Feb 2012 Posts: 8 Thanks: 0 Combination containing a pair I've just been looking and permutations and combinations in a B&C textbook and one of the following exercises stumped me. Q. There's several of each of nine items. How many combinations are there that contain 2 of the same item and two other different items. The permutations for all different items is 9C4 = 9!/(5!4!) = 126 and all 4 the same would be 9. The answer given for the Q is 252 and I can't see how they get this. My solution would be that initially you have 9 choices then the second of the same item is a given and then what's left is the number of combinations of 2 from 7. Soln: 9 * 1 * (7!/(5!/2!)) = 189 (ie. not the answer given 252). What have I missed?
 January 13th, 2016, 06:20 PM #2 Global Moderator   Joined: May 2007 Posts: 6,234 Thanks: 496 Wrong forum - try statistics
 January 28th, 2016, 12:08 PM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 2,480 Thanks: 627 But certainly, just saying "There's several of each of nine items" is not enough. How many of the individual items? That will affect the answer.
 January 28th, 2016, 09:02 PM #4 Global Moderator   Joined: May 2007 Posts: 6,234 Thanks: 496 I remember seeing the same problem in another forum with the same mistake. To get 252 use $\displaystyle 9\frac{8!}{6!2!}$

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