The number of nonnegative integer solutions of the equation `x+y+z+5t = 15` is

Options

A. 196
B. 224
C. 312
D. 364

Answer

To find the number of non-negative integer solutions for the equation \( x + y + z + 5t = 15 \), we can break down the problem step by step.

### Step 1: Understand the equation
We have the equation:
\[ x + y + z + 5t = 15 \]
where \( x, y, z, \) and \( t \) are non-negative integers.

### Step 2: Analyze the variable \( t \)
Since \( t \) is multiplied by 5, we can consider different values for \( t \) and find the corresponding values for \( x + y + z \).

### Step 3: Calculate for different values of \( t \)

1. **For \( t = 0 \):**
\[ x + y + z = 15 \]
The number of non-negative integer solutions is given by the formula:
\[ \binom{n + r - 1}{r - 1} \]
Here, \( n = 15 \) and \( r = 3 \) (for \( x, y, z \)):
\[ \text{Number of solutions} = \binom{15 + 3 - 1}{3 - 1} = \binom{17}{2} = \frac{17 \times 16}{2 \times 1} = 136 \]

2. **For \( t = 1 \):**
\[ x + y + z = 15 - 5 = 10 \]
The number of solutions is:
\[ \text{Number of solutions} = \binom{10 + 3 - 1}{3 - 1} = \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66 \]

3. **For \( t = 2 \):**
\[ x + y + z = 15 - 10 = 5 \]
The number of solutions is:
\[ \text{Number of solutions} = \binom{5 + 3 - 1}{3 - 1} = \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]

4. **For \( t = 3 \):**
\[ x + y + z = 15 - 15 = 0 \]
The number of solutions is:
\[ \text{Number of solutions} = \binom{0 + 3 - 1}{3 - 1} = \binom{2}{2} = 1 \]

### Step 4: Sum the solutions
Now, we sum the number of solutions for all values of \( t \):
\[ 136 + 66 + 21 + 1 = 224 \]

### Final Answer
The total number of non-negative integer solutions for the equation \( x + y + z + 5t = 15 \) is:
\[ \boxed{224} \]