In How Many Different Ways can 3 Identical Green Shirts and 3 Identical Red Shirts be Distributed Among 6 Children GMAT Problem Solving

Question: 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?

1. 20
2. 40
3. 216
4. 720
5. 729

“In How Many Different Ways can 3 Identical Green Shirts and 3 Identical Red Shirts be Distributed Among 6 Children GMAT Problem Solving” is a question of GMAT Quantitative reasoning section. This question has been taken from the book "GMAT Prep Course" published in the year 2004. GMAT Quant section consists of a total of 31 questions. This is a GMAT Data Sufficiency question that allows the candidates to prove if the statement is sufficient as per the options. The total time allotted for this part is 62 minutes which allows the candidate 2 minutes to answer each question.

Solution and Explanation

Approach Solution 1:

Explanation: it is asked 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?

This question can be simplified to: How many different methods are there to select three of the six kids to receive a green shirt?
Keep in mind that after we have given each of those 3 selected children a green shirt, the other children have to get red shirts. In other words, the children who receive red shirts are locked until we have given green shirts to 3 children.

How many different methods are there to choose three of the six children to get green shirts?
This is a combination question because it does not matter what order the chosen children are in.

From a pool of six children, we can select three in 6C3 (or 20) different ways.

The answer is A which is 20.

Correct Answer: A

Approach Solution 2:

Explanation: There is another approach to solve the question.

it is asked 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?

If we use the letters G and R to represent a green shirt and a red shirt, the issue becomes how to arrange 3 Gs and 3 Rs in the string GGGRRR. The following formula, which applies the idea of permutations with the repetition of indistinguishable items, provides the solution:

\(P = \frac{N!}{(r1!)*(r2!)*(r3!)*.....*(rn!)}\)

N stands for the total number of objects that need to be ordered in this formula. Each of the I indistinguishable objects' frequencies is represented by a ri (i = 1, 2, 3, …, n)

The frequency merely refers to how frequently the identical item appears in the set. Take note that there are 3 identical (indistinguishable) green shirts and 3 identical red shirts (indistinguishable).

We can arrange three Gs and three Rs in the string GGGRRR in the following ways:

6!/(3! × 3!)
= 720/(6 × 6)
= 720/36
= 20

The answer is A which is 20.

Correct Answer: A

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