Let A = Σᵢ₌₁¹⁰ Σⱼ₌₁¹⁰ min { i,j} and B = Σᵢ₌₁¹⁰ Σⱼ₌₁¹⁰ max { i,j} . Then A + B is equal to _____________.

Solution

Maths sir · 4 steps

Do not touch min and max separately: for any pair, the smaller plus the larger is just i + j.

1. Pair them up. Add the two double sums term by term. For every pair (i, j), one of the two numbers is the minimum and the other is the maximum, so min{i,j} + max{i,j} = i + j. This holds even when i = j.

   A + B = Σᵢ₌₁¹⁰Σⱼ₌₁¹⁰ (i + j)

2. Split the sum. For each fixed j, the term i runs over 1 to 10, so ΣΣ i = 10 × 55 = 550. In the same way, ΣΣ j = 10 × 55 = 550.
3. Quick check. Separately, A = 1² + 2² + … + 10² = 385 and B = 715. They add to the same total.
4. So. A + B = 550 + 550 = 1100.

The common slip: 550. Summing i + j over only one index forgets the other 10 values of the second index. That gives 55 + 55 × 10 or just 10 × 55 = 550, instead of 1100.

The idea to keep. When a question asks for the sum of min and max together, use min + max = i + j. A double sum over an n × n grid is then 2n × n(n+1)/2.

Answer from the NTA final answer key. Solution by PrepWiser.