How many different pairs (x, y) can be formed using the numbers {1, 2, 3,..., 43) such that x < y and the sum of x and y is even?

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  1. 420
  2. 451
  3. 441
  4. 460

Answer (Detailed Solution Below)

Option 3 : 441
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Detailed Solution

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Given:

Set of numbers = {1, 2, 3, ..., 43}

x + y is even

Formula used:

Total pairs where x < y = nC2 = \(\frac{n(n-1)}{2}\)

Even pairs condition: x + y is even ⇒ x and y both even or both odd

Calculation:

Total even numbers = 22

Total odd numbers = 21

Pairs from even numbers = \(\frac{22×21}{2}\)

⇒ Pairs = 231

Pairs from odd numbers = \(\frac{21×20}{2}\)

⇒ Pairs = 210

Total pairs = 231 + 210

⇒ Total = 441

∴ The correct answer is option (3).

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