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CAT 2017 Slot 1 — QA Question 25

Fundamental Principles of P&CEasy

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

Answer & solution

Correct answer: 99

  • A

    101

  • 99

  • C

    87

  • D

    105

Solution

Easy

Solve for xx to bound it, then count the integer pairs (y,z)(y,z) that fit. Each valid (y,z)(y,z) fixes x=25+y+zx=25+y+z uniquely, so counting pairs counts solutions.

1

Express xx and find its range. From the equation x=25+y+zx = 25+y+z. Since y,z≥1y,z\ge 1, the smallest xx is 2727; and x≤40x\le 40 is given.

x=25+y+z⇒ 27≤x≤40(y,z≥1 and x≤40)⇒ y+z≤15(x=25+y+z≤40)\begin{aligned} &x = 25 + y + z\\ &\Rightarrow\ 27 \le x \le 40 \quad\text{(}y,z\ge 1\text{ and }x\le 40\text{)}\\ &\Rightarrow\ y + z \le 15 \quad\text{(}x=25+y+z\le 40\text{)} \end{aligned}
2

Count integer pairs (y,z)(y,z) with 1≤y,z≤121\le y,z\le 12 and y+z≤15y+z\le 15. For a fixed sum ss, the number of pairs in [1,12]2[1,12]^2 is s−1s-1 when s≤13s\le 13 and 25−s25-s when s≥14s\ge 14.

∑s=213(s−1)=1+2+⋯+12=78(sums s≤13)s=14: 25−14=11s=15: 25−15=10\begin{aligned} &\textstyle\sum_{s=2}^{13}(s-1) = 1+2+\cdots+12 = 78 \quad\text{(sums }s\le 13\text{)}\\ &s=14:\ 25-14 = 11\\ &s=15:\ 25-15 = 10 \end{aligned}
3

Add up. Total number of triples (x,y,z)(x,y,z).

78+11+10=99\begin{aligned} &78 + 11 + 10 = 99 \end{aligned}
99 solutions99 \text{ solutions}

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CAT 2017 Slot 1 QA Q25: The number of solutions (x, y, z) to the equation x – y – z = 25, where x, y, and z are positive i — Solution | TheCATExam