Σₖ₌₀⁶ ⁵¹⁻ᵏC₃ is equal to :

Solution

Maths sir · 4 steps

Write each ⁿC₃ as a difference of two ⁿC₄ terms, and the whole sum telescopes.

1. List the terms. As k runs from 0 to 6, the sum is ⁵¹C₃ + ⁵⁰C₃ + … + ⁴⁵C₃. That is seven terms.
2. Use Pascal's rule. Pascal's rule says ⁿC₃ + ⁿC₄ = ⁿ⁺¹C₄. Rearranged, each term becomes a difference of two ⁿC₄ values.

   ⁿC₃ = ⁿ⁺¹C₄ - ⁿC₄

3. Telescope. Write this out for n = 51, 50, …, 45: you get (⁵²C₄ - ⁵¹C₄) + (⁵¹C₄ - ⁵⁰C₄) + … + (⁴⁶C₄ - ⁴⁵C₄). Every middle term cancels with its neighbour.
4. So. Only the first and last terms survive, so the sum is ⁵²C₄ - ⁴⁵C₄, which is option (d).

The idea to keep. A sum of ⁿCᵣ with the same r and consecutive n always telescopes: the lower index goes up by one, and the upper index goes one past the largest term.

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