(\sum\_{r = 0}^{50}{100{C\_{r}}^{200}C\_{150 + r}}) = | MATH - MATHEMATICAL INDUCTION AND DIVISIBILITY PROBLEMS | SolveeIt

$\sum_{r = 0}^{50}{100{C_{r}}^{200}C_{150 + r}}$ =

³⁰⁰C₅₀

¹⁰⁰C₅₀

¹⁰⁰C₅₀ ²⁰⁰C₅₀

None of these

Answer

³⁰⁰C₅₀

Explanation

$\sum_{r = 0}^{50}{100{C_{r}}^{200}C_{150 + r}}$ = ¹⁰⁰C₀ ²⁰⁰C₁₅₀ + ¹⁰⁰C₁²⁰⁰C₁₅₁ + ¹⁰⁰C₂ ²⁰⁰C₁₅₂ +….+¹⁰⁰C₅₀ ²⁰⁰C₂₀₀.

= coefficient of x⁵⁰ in (1 + x)¹⁰⁰ (x + 1)²⁰⁰

= (1 + x)³⁰⁰ is ³⁰⁰C₅₀

Question: \(\sum_{r = 0}^{50}{100{C_{r}}^{200}C_{150 + r}}\) =...