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In how many different ways can 3 identical green shirts and
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Updated on: 24 Feb 2012, 22:58

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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?

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: In how many different ways can 3 identical green shirts and
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24 Feb 2012, 22:59

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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?

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).

Re: In how many different ways can 3 identical green shirts and
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30 Mar 2015, 01:55

Temurkhon wrote:

What if we distribute 6 shirts among 4 children?

R-R-R-G-G-G (shirts) 1-2-3-4 (children)

RRRG RRGG RGGG GGGR GGRR GRRR RGRG GRGR RGGR GRRG

Looks like 10

algebraically 6*5*4*3/3! *3!=10

Is that right?

You mean each child gets exactly one shirt?

You have missed 4 cases: GRGG, GGRG, RGRR, RRGR

There will be total 14 cases. Say, had there been 4 of each type of shirt, each child could have got a shirt in two ways: Red or Green. This would give us 2*2*2*2 = 16 ways. But "All 4 children get Red" and "All 4 children get Green" are two cases which are not possible. So total cases are 16 - 2 = 14
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Re: In how many different ways can 3 identical green shirts and
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29 Aug 2016, 14:14

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Top Contributor

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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?

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

We can take this question and ask an easier question: In how many ways can we choose 3 of the 6 children to receive a green shirt?

Notice that, once we have given a green shirt to each of those 3 chosen children, the REMAINING 3 children must get red shirts. In other words, once we have given green shirts to 3 children, the children who get red shirts is locked.

So, in how many ways can we select 3 of the 6 children to receive a green shirt? Since the order of the selected children does not matter, this is a combination question. We can choose 3 children from 6 children in 6C3 ways (= 20 ways)

Re: In how many different ways can 3 identical green shirts and
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29 Aug 2016, 19:15

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

6! / 3!x3! = 120/6= 20 ways
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Re: In how many different ways can 3 identical green shirts and
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29 Nov 2016, 16:28

1

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

If we let G denote a green shirt and R denote a red shirt, the problem becomes how to arrange 3 Gs and 3 Rs in the string of GGGRRR. The answer can be found using the concept of permutations with repetition of indistinguishable objects, using the following formula:

In this formula, N represents the total number of objects to be arranged. Each ri (i = 1, 2, 3, …, n) represents the frequency of each of the i indistinguishable objects.

The frequency simply means the number of times that the indistinguishable item occurs in the set. Note that there are 3 green shirts that are identical (indistinguishable), and there are 3 red shirts that are identical (indistinguishable).

Therefore, the number of ways we can arrange 3 Gs and 3 Rs in the string of GGGRRR is:

Re: In how many different ways can 3 identical green shirts and
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13 Feb 2018, 22:58

Hi All,

If we had 6 different shirts, then there would be 6! = 720 options. However, since there are 3 IDENTICAL red shirts and 3 IDENTICAL green shirts, we have to do some extra work on our calculations.

If we call 3 of the children A, B and C and each of them gets an identical red shirt, then there are technically 6 different ways for those 3 shirts to be given to those 3 children. Since the shirts are identical though, we are NOT supposed to count this as 6 different options....it should only be counted as 1 option.

Thus, we would have to divide 720 by 6....

We would have to do the same thing with the identical green shirts, which means we'd have to divide by 6 AGAIN.

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