Bodhee Prep-CAT Online Preparation

CAT 2017 [slot 1] Question with solution 25

Question 25:
The number of solutions ( x, y, z) to the equation x - y - z = 25, where x, y, and z are positive integers such that $x\leq40,y\leq12$, and $z\leq12$ is
  1. 101
  2. 99
  3. 87
  4. 105
Option: 2
Explanation:

x - y - z = 25 and $x\leq40,y\leq12$, $z\leq12$
If x = 40 then y + z = 15. Now since both y and z are natural numbers less than 12, so y can range from 3 to 12 giving us a total of 10 solutions.Similarly, if x = 39, then y + z = 14. Now y can range from 2 to 12 giving us a total of 11 solutions.
If x = 38, then y + z = 13. Now y can range from 1 to 12 giving us a total of 12 solutions.
If x = 37 then y + z = 12 which will give 11 solutions.
Similarly on proceeding in the same manner the number of solutions will be 10, 9, 8, 7 and so on till 1.
Hence, required number of solutions will be (1 + 2 + 3 + 4 . . . . + 12) + 10 + 11
= 12×13/2 + 21
78 + 21 = 99


Previous QuestionNext Question

CAT 2023
Classroom Course

We are starting classroom course for CAT 2023 in Gurugram from the month of December.
Please fill the form to book your seat for FREE Demo Classes

CAT 2023 Classroom Course starts in Gurgaon