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Number of Combinations

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Number of Combinations

by mainhoon » Sun Aug 15, 2010 7:29 pm
A B C
D E F
G H I

So I have these 9 objects above. Objective to get 4 objects as long as 1 is selected from each row.

- I want to select 2 from a row and then 1 each from the other 2. The # is then 3 3C2 3C1 3C1 = 81.

- But here is my question. I select 1 from each row 1st and then select the 4th one from the 6 remaining. That # is 3C1 3C1 3C1 6C1 = 27 x 6 = 162. Clearly double the # as above. How can I rationalize? Where is the double counting?
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by vijaynaik » Sun Aug 15, 2010 9:49 pm
I think conceptually 2nd example is incorrect because in the 1st example each row was considered as independent entity to pick from.

lets say there r 2 lines (each line has thousands of people) and we need to pick one person from either of the line. How many ways can we do that? i would say 2C1. Pick 1 from 2 lines it doesn't matter how many are there in the lines.

so it should be 3C1 * 3C1 * 3C1 * ( 3C1 -- 1 object from any of the 3 rows).

I may be wrong :-)
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by mainhoon » Sun Aug 15, 2010 9:52 pm
The right answer is 81. So option 1 is correct. I want to know how to get the factor of 2 in option 2 to reduce to option 1.
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by vijaynaik » Sun Aug 15, 2010 10:22 pm
in the 2nd option it should not be 6c1 but it should be 3c1. That gives 81. Why 3c1? because we have to pick 1 from any of the 3 rows not 6. sorry i couldn't explain it any better.
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by Prashantbhardwaj » Sun Aug 15, 2010 11:07 pm
Well you have to divide 6C1 in the option2 with 2 because there are 3 pairs of indistinguishable objects. (2 belonging two same row), hence the selections possible will be half.
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