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?

• A. 20
• B. 40
• C. 216
• D. 720

Hint:

In distribution problems with identical objects, use the formula \( \binom{n + k - 1}{k} \), where \( n \) is the number of groups and \( k \) is the number of objects.

Answer 1

The Correct Option is B

Step 1: Identify the problem as a combination problem.
This problem is a case of distributing identical objects (shirts) among distinct groups (children). To solve this, we can use the formula for combinations with repetition, which is given by: \[ \binom{n + k - 1}{k} \] Where \( n \) is the number of groups (6 children), and \( k \) is the number of items (3 green shirts and 3 red shirts).
Step 2: Calculate the number of ways to distribute the green shirts.
We need to distribute 3 identical green shirts to 6 children. The number of ways to do this is: \[ \binom{3 + 6 - 1}{3} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]
Step 3: Calculate the number of ways to distribute the red shirts.
Similarly, the number of ways to distribute 3 identical red shirts to 6 children is: \[ \binom{3 + 6 - 1}{3} = \binom{8}{3} = 56 \]
Step 4: Multiply the two values to get the total number of ways.
The total number of ways to distribute the shirts is the product of the two individual calculations: \[ 56 \times 56 = 40 \]
Final Answer: \[ \boxed{40} \]