Question

The number of solutions $(x, y, z)$ to the equation $x\ –\ y \ –\ z = 25$, where x, y, and z are positive integers such that $x ≤ 40, y ≤ 12$, and $z ≤ 12$ is

• A. 101
• B. 99
• C. 87
• D. 105

Answer 1

Answer: (B)

99

Answer 2

The equation $x−y−z=25$ can be expressed as $x=25+y+z.$
Given that y and z are positive integers with $y≤12$ and $z≤12$, the range for $y+z$ is $2≤(x+y)≤15$ when $27≤x≤40.$
The minimum value for x is 27.
For $y=1, z$ can take 12 values.
Similarly, for $y=2, z$ can take 12 values, and so on, until $y=12$ where z can take 10 values.
Therefore, the total number of solutions is $3+4+5+6+7+8+9+10+11+12+12+12=99.$
Hence, the required result is $99.$