Correct option is B
Given:
Three positive integers a, b, c such that:
a + b + c = 15
We are to find the minimum value of the expression:
(a − 2)² + (b − 2)² + (c − 2)²
Concept:
To minimize the sum of squares, the numbers should be as close to each other as possible.
Among all positive integers adding up to a fixed sum, the minimum sum of squares occurs when the numbers are equal or nearly equal.
Solution:
Try a = b = c = 5
This satisfies: a + b + c = 5 + 5 + 5 = 15 ️
Now calculate:
(5 − 2)² + (5 − 2)² + (5 − 2)²
= 3² + 3² + 3²
= 9 + 9 + 9 = 27
Now test another case to compare:
Try consecutive numbers like a = 4, b = 5, c = 6
→ Sum = 15
→ Expression = (4 − 2)² + (5 − 2)² + (6 − 2)² = 4 + 9 + 16 = 29
So, 27 is the minimum.
Final Answer: (B) 27
