A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

Answer

To solve the problem of selecting 2 black balls from a total of 5 black balls and 3 red balls from a total of 6 red balls, we can use the concept of combinations. Here’s a step-by-step solution:

### Step 1: Determine the number of ways to select 2 black balls from 5 black balls.
We use the combination formula, which is given by:
\[ nCk = \frac{n!}{k!(n-k)!} \]
For selecting 2 black balls from 5, we have:
\[ 5C2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} \]
Calculating this:
\[ 5C2 = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10 \]

### Step 2: Determine the number of ways to select 3 red balls from 6 red balls.
Using the same combination formula, for selecting 3 red balls from 6, we have:
\[ 6C3 = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!} \]
Calculating this:
\[ 6C3 = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{120}{6} = 20 \]

### Step 3: Combine the results from Step 1 and Step 2.
The total number of ways to select 2 black balls and 3 red balls is the product of the two combinations calculated:
\[ \text{Total Ways} = 5C2 \times 6C3 = 10 \times 20 = 200 \]

### Final Answer:
Thus, the number of ways to select 2 black balls and 3 red balls is **200**.

---