How many triplets (x, y, z) satisfy the equation x + y + z = 6, where x, y and z are natural numbers ?
- (a) 4
- (b) 5
- (c) 9
- (d) 10 ✓ UPSC's answer
Why the answer is (d)
• Let x, y, z be natural numbers, i.e., positive integers at least 1.
• Put x' = x - 1, y' = y - 1, z' = z - 1, so x', y', z' are non-negative integers.
• The equation becomes x' + y' + z' = 6 - 3 = 3.
• The number of non-negative integer solutions to x' + y' + z' = 3 is C(3 + 3 - 1, 3 - 1) = C(5, 2) = 10.
• Hence the number of triplets (x, y, z) is 10, which is option (d).
Why the other options are wrong
- (a) 4
- Option (a) gives 4, but the stars-and-bars count for positive integer solutions is C(5, 2) = 10.
- (b) 5
- Option (b) gives 5, which is not the number of positive integer solutions to x + y + z = 6.
- (c) 9
- Option (c) gives 9, but the correct count is C(5, 2) = 10.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.