Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is

Options

A. 200
B. 300
C. 500
D. 350

Answer

To solve the problem of forming teams consisting of 2 girls and 3 boys from a class of 5 girls and 7 boys, where two specific boys A and B refuse to be on the same team, we can follow these steps:

### Step 1: Calculate the total number of teams without restrictions
First, we need to find the total number of ways to choose 2 girls from 5 and 3 boys from 7.

- The number of ways to choose 2 girls from 5 is given by the combination formula:
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

- The number of ways to choose 3 boys from 7 is:
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

Now, we multiply the number of ways to choose the girls and boys:
\[ \text{Total teams} = \binom{5}{2} \times \binom{7}{3} = 10 \times 35 = 350 \]

### Step 2: Calculate the number of teams where A and B are together
Next, we need to find the number of teams where boys A and B are both included. If A and B are in the team, we need to choose 1 more boy from the remaining 5 boys and 2 girls from the 5 girls.

- The number of ways to choose 2 girls from 5 remains:
\[ \binom{5}{2} = 10 \]

- The number of ways to choose 1 boy from the remaining 5 boys is:
\[ \binom{5}{1} = 5 \]

Now, we multiply the number of ways to choose the girls and the additional boy:
\[ \text{Teams with A and B together} = \binom{5}{2} \times \binom{5}{1} = 10 \times 5 = 50 \]

### Step 3: Calculate the number of valid teams
Finally, we subtract the number of teams where A and B are together from the total number of teams to find the number of teams where A and B are not together:
\[ \text{Valid teams} = \text{Total teams} - \text{Teams with A and B together} = 350 - 50 = 300 \]

### Final Answer
The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, ensuring that boys A and B are not on the same team, is **300**.

---