In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Options

A. `.^9C_6`
B. `.^3C_3*.^5C_3`
C. `.^3C_5*.^3C_4`
D. `.^5C_3*.^4C_3`

Explanation

To solve the problem of selecting a team of 3 boys and 3 girls from 5 boys and 4 girls, we can use the concept of combinations. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the number of ways to select boys
We need to select 3 boys from a total of 5 boys. The number of ways to choose 3 boys from 5 can be calculated using the combination formula:

\[ \text{Number of ways to choose 3 boys} = \binom{5}{3} \]

Using the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\):

\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 2: Determine the number of ways to select girls
Next, we need to select 3 girls from a total of 4 girls. The number of ways to choose 3 girls from 4 can be calculated similarly:

\[ \text{Number of ways to choose 3 girls} = \binom{4}{3} \]

Using the combination formula:

\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = \frac{4}{1} = 4 \]

### Step 3: Calculate the total number of ways to form the team
To find the total number of ways to select the team of 3 boys and 3 girls, we multiply the number of ways to choose the boys by the number of ways to choose the girls:

\[ \text{Total ways} = \binom{5}{3} \times \binom{4}{3} = 10 \times 4 = 40 \]

### Final Answer
Thus, the total number of ways to select a team of 3 boys and 3 girls from 5 boys and 4 girls is **40**.

---