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
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
Updated On: Sep 26, 2024
- 101
- 99
- 87
- 105
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The Correct Option is B
Solution and Explanation
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 \(99.\)
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