Combinations Equation

Solving an equation involving a single combination reduces to a quadratic in n, a routine JEE Advanced algebraic step. The relation C(n, 2) = n(n - 1)/2 = 28 gives n(n - 1) = 56. Seeking two consecutive integers with product 56, we find 8 × 7 = 56, so n = 8. Equivalently, n^2 - n - 56 = 0 factors as (n - 8)(n + 7) = 0, whose positive root is n = 8. Option 7 gives C(7,2) = 21, too small. Option 14 gives C(14,2) = 91, too large. Option 9 gives C(9,2) = 36, also incorrect. Hence n = 8. Plausibility check: substituting n = 8 yields C(8,2) = 8·7/2 = 28, exactly the required value, confirming the positive integer solution of the combination equation. The block method for togetherness conditions ties the constrained items into a single unit, arranges the resulting units, and then multiplies by the internal arrangements of the block. Its natural counterpart is the gap method for separation conditions, and recognizing which of these complementary techniques a problem demands is the central strategic decision in arrangement counting.