Question

In how many ways we can select a group of 2 males and 2 females from the family of 5 males and 3 females?

• A. \( 3 \)
• B. \( 30 \)
• C. \( 20 \)
• D. None of the above

Hint:

When selecting items for a group where the order doesn't matter, use combinations (\( C(n, k) \)). If selections are made from different categories (like males and females) to form one group, multiply the number of ways for each category.

Answer 1

The Correct Option is B

Step 1: Identify the number of males and females available and the number to be selected. Available males = 5, Males to select = 2. Available females = 3, Females to select = 2.

Step 2: Calculate the number of ways to select the males. This is a combination problem since the order of selection does not matter. The number of ways to choose 2 males from 5 is given by the combination formula \( C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!} \). \[ \text{Ways to select males} = C(5, 2) = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{5 \times 4}{2} = 10 \] Step 3: Calculate the number of ways to select the females. The number of ways to choose 2 females from 3 is: \[ \text{Ways to select females} = C(3, 2) = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3 \times 2!}{2! \times 1} = 3 \] Step 4: Calculate the total number of ways to select the group. Since the selection of males and females are independent events, we multiply the number of ways for each selection (using the multiplication principle). \[ \text{Total ways} = (\text{Ways to select males}) \times (\text{Ways to select females}) \] \[ \text{Total ways} = 10 \times 3 = 30 \] Step 5: Compare the result with the given options. The calculated number of ways is 30, which matches option (B).