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# In how many different ways can 3 identical green shirts and

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In how many different ways can 3 identical green shirts and [#permalink]  25 Nov 2010, 18:54
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In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?

A. 20
B. 40
C. 216
D. 720
E. 729
[Reveal] Spoiler: OA

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Last edited by Bunuel on 24 Feb 2012, 22:58, edited 1 time in total.
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Re: Combination [#permalink]  25 Nov 2010, 19:51
GGG RRR

Therefore total number of ways is

6! but there are two groups of 3 identical things.

Therefore total number of "different" ways is

6!/ (3!) (3!) = 20
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Re: Combination [#permalink]  25 Nov 2010, 20:01
No of ways 6 shirts can be distributed among 6 people 6!
Since 3 red are identical and 3 green are identical = (6!)/3!*3!=20
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Re: Combination [#permalink]  25 Nov 2010, 21:43
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shrive555 wrote:
In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?

20
40
216
720
729

Or out of 6 children, choose 3 in 6C3 ways = 20 ways.

Note: When you choose 3 children say, A, B and C are give them a red shirt, D, E and F get a green shirt. When you choose D, E and F and give them a red shirt, A, B and C automatically get the green shirts. So you do not need to multiply by 2! above.
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Re: Combination [#permalink]  24 Feb 2012, 17:19
Approach 1:
1st Child: 6 has options
2nd Child: 5 has options…
Therefore, for all kids: 6 x 5 x 4 x 3 x 2 = 720 arrangements.

Since the reds are identical, we divide by 3!; Since the greens are identical, we divide by another 3!

So: in all, 720/[ 3! X 3! ] = 20 ways.

Approach 2 / MGMAT technique:
This is like anagramming RRRGGG. No of arrangements = 6! / 3! x 3! ways ==> 20.
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Re: In how many different ways can 3 identical green shirts and [#permalink]  24 Feb 2012, 22:59
shrive555 wrote:
In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?

A. 20
B. 40
C. 216
D. 720
E. 729

1-2-3-4-5-6 (children)
B-B-B-G-G-G (shirts)
G-B-B-G-G-B
G-G-B-G-B-B
....

So, basically # of assignments of 6 shirts to 6 children (such that each child receives a shirt) equals to # of permutations of 6 letters BBBGGG, which is \frac{6!}{3!3!}=20 (we divide by 3!*3!, since there are 3 identical B's and 3 identical G's).

Hope it helps.
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Re: In how many different ways can 3 identical green shirts and   [#permalink] 24 Feb 2012, 22:59
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