Question

The value of `""^50"C"_4 + sum_("r" = 1)^6 ""^(56 - "r")"C"_3` is ______.

पर्याय

• ⁵⁵C₄

• ⁵⁵C₃

• ⁵⁶C₃

• ⁵⁶C₄

Answer 1

The value of `""^50"C"_4 + sum_("r" = 1)^6 ""^(56 - "r")"C"_3` is ⁵⁶C₄.

Explanation:

We need to find value of `""^50"C"_4 ​+ sum_("r" = 1)^6 ""^(​56 − "r")"C"_3`​

= ⁵⁰C₄ + ⁵⁰C₃ + ⁵¹C₃ + ⁵²C₃ + ⁵³C₃ + ⁵⁴C₃ + ⁵⁵C₃

(∵ ⁿC_r + ⁿC_r–1 = ⁿ⁺¹C₁)

= ⁵¹C₄ + ⁵¹C₃ + ⁵²C₃ + ⁵³C₃ + ⁵⁴C₃ + ⁵⁵C₃

= ⁵²C₄ + ⁵²C₃ + ⁵³C₃ + ⁵⁴C₃ + ⁵⁵C₃

= ⁵³C₄ + ⁵³C₃ + ⁵⁴C₃ + ⁵⁵C₃

= ⁵⁴C₄ + ⁵⁴C₃ + ⁵⁵C₃

= ⁵⁵C₄ + ⁵⁵C₃

= ⁵⁶C₄