Question

The number of triplets \( (x, y, z) \), where \( x, y, z \) are distinct non-negative integers satisfying \( x + y + z = 15 \), is:

• A. 136
• B. 114
• C. 80
• D. 92

Hint:

For problems involving distinct non-negative integer solutions, be sure to consider the possibility of equal values for the variables, and subtract those cases from the total number of solutions.

Answer 1

The Correct Option is B

We are given that \( x + y + z = 15 \) and we are asked to find the number of distinct non-negative integer solutions to this equation.
The total number of non-negative integer solutions to the equation \( x + y + z = 15 \) is given by the formula: \[ \text{Total number of solutions} = \binom{15 + 3 - 1}{3 - 1} = \binom{17}{2} = 136 \] This is the total number of solutions without considering whether the values of \( x \), \( y \), and \( z \) are distinct or not.
Now, to find the number of distinct solutions, let's consider the case when \( x = y = z \).
Let \( x = y = z \). Then, \[ x + x + x = 15 \quad \Rightarrow \quad 3x = 15 \quad \Rightarrow \quad x = 5 \] Thus, there is exactly 1 solution where \( x = y = z = 5 \).
Next, we need to account for the cases where two of \( x \), \( y \), and \( z \) are equal. Suppose \( x = y \), then: \[ x + x + z = 15 \quad \Rightarrow \quad 2x + z = 15 \] Solving for \( z \), we get: \[ z = 15 - 2x \] We require that \( x \neq z \), so the distinct solutions occur when \( x \) takes values from 1 to 7. For each value of \( x \), there is exactly one value for \( z \) that satisfies the equation. Therefore, there are 7 solutions where two of \( x \), \( y \), and \( z \) are equal.
Thus, the total number of distinct solutions is: \[ 136 - 1 - 7 = 114 \] Therefore, the number of distinct non-negative integer triplets \( (x, y, z) \) satisfying \( x + y + z = 15 \) is \( 114 \).