The number of ways in which 10 balls can be selected from 10 identical green balls, 10 identical blue balls and 9 idenitcal red balls are

Options

A. 63
B. 64
C. 65
D. 66

Answer

To solve the problem of selecting 10 balls from 10 identical green balls, 10 identical blue balls, and 9 identical red balls, we can use the concept of combinations with restrictions. We will denote the number of green balls selected as \( g \), the number of blue balls as \( b \), and the number of red balls as \( r \). The equation we need to satisfy is:

\[ g + b + r = 10 \]

with the constraints:
- \( 0 \leq g \leq 10 \)
- \( 0 \leq b \leq 10 \)
- \( 0 \leq r \leq 9 \)

### Step 1: Analyze the cases based on the number of red balls selected

We will consider different cases based on the number of red balls \( r \) selected, since there are only 9 red balls available.

#### Case 1: \( r = 0 \)
In this case, we have:
\[ g + b = 10 \]
The number of non-negative integer solutions is given by:
\[ \text{Number of solutions} = 11 \quad (\text{from } g = 0 \text{ to } g = 10) \]

#### Case 2: \( r = 1 \)
Here, we have:
\[ g + b = 9 \]
The number of solutions is:
\[ 10 \quad (\text{from } g = 0 \text{ to } g = 9) \]

#### Case 3: \( r = 2 \)
Now, we have:
\[ g + b = 8 \]
The number of solutions is:
\[ 9 \quad (\text{from } g = 0 \text{ to } g = 8) \]

#### Case 4: \( r = 3 \)
We have:
\[ g + b = 7 \]
The number of solutions is:
\[ 8 \quad (\text{from } g = 0 \text{ to } g = 7) \]

#### Case 5: \( r = 4 \)
We have:
\[ g + b = 6 \]
The number of solutions is:
\[ 7 \quad (\text{from } g = 0 \text{ to } g = 6) \]

#### Case 6: \( r = 5 \)
We have:
\[ g + b = 5 \]
The number of solutions is:
\[ 6 \quad (\text{from } g = 0 \text{ to } g = 5) \]

#### Case 7: \( r = 6 \)
We have:
\[ g + b = 4 \]
The number of solutions is:
\[ 5 \quad (\text{from } g = 0 \text{ to } g = 4) \]

#### Case 8: \( r = 7 \)
We have:
\[ g + b = 3 \]
The number of solutions is:
\[ 4 \quad (\text{from } g = 0 \text{ to } g = 3) \]

#### Case 9: \( r = 8 \)
We have:
\[ g + b = 2 \]
The number of solutions is:
\[ 3 \quad (\text{from } g = 0 \text{ to } g = 2) \]

#### Case 10: \( r = 9 \)
We have:
\[ g + b = 1 \]
The number of solutions is:
\[ 2 \quad (\text{from } g = 0 \text{ to } g = 1) \]

### Step 2: Summing the solutions

Now, we sum all the solutions from each case:
\[ 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 = 55 \]

### Final Answer
Thus, the total number of ways to select 10 balls from the given balls is:

\[ \boxed{55} \]