The number of non negative integral solution of the equation, `x+ y+3z =33` is

Options

A. 120
B. 135
C. 210
D. 520

Answer

To find the number of non-negative integral solutions of the equation \(x + y + 3z = 33\), we can follow these steps:

### Step 1: Rewrite the equation
We can express the equation in terms of \(x\) and \(y\):
\[ x + y = 33 - 3z \]
This indicates that for each non-negative integer value of \(z\), \(x + y\) must also be a non-negative integer.

### Step 2: Determine the range of \(z\)
Since \(x\) and \(y\) must be non-negative, the right-hand side of the equation \(33 - 3z\) must be non-negative:
\[ 33 - 3z \geq 0 \implies z \leq 11 \]
Thus, \(z\) can take values from \(0\) to \(11\) (inclusive).

### Step 3: Calculate the number of solutions for each \(z\)
For each fixed value of \(z\), we can determine the number of non-negative integral solutions for \(x + y = 33 - 3z\). The number of non-negative integral solutions to the equation \(x + y = n\) is given by the formula:
\[ \text{Number of solutions} = n + 1 \]
Thus, for each \(z\), the number of solutions is:
\[ 33 - 3z + 1 = 34 - 3z \]

### Step 4: Sum the solutions for all valid \(z\)
Now we need to sum the number of solutions for all values of \(z\) from \(0\) to \(11\):
\[ \text{Total solutions} = \sum_{z=0}^{11} (34 - 3z) \]

### Step 5: Calculate the total
This can be simplified as follows:
\[ \text{Total solutions} = \sum_{z=0}^{11} 34 - \sum_{z=0}^{11} 3z \]
Calculating each part:
1. The first part:
\[ \sum_{z=0}^{11} 34 = 34 \times 12 = 408 \]
2. The second part (using the formula for the sum of the first \(n\) integers):
\[ \sum_{z=0}^{11} z = \frac{11(11 + 1)}{2} = \frac{11 \times 12}{2} = 66 \]
Therefore,
\[ \sum_{z=0}^{11} 3z = 3 \times 66 = 198 \]

Putting it all together:
\[ \text{Total solutions} = 408 - 198 = 210 \]

### Final Answer
Thus, the number of non-negative integral solutions of the equation \(x + y + 3z = 33\) is \(210\).

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