Question

Prove that C(50,3) + C(50,4) = C(51,4)

Answer 1

By definition of the binomial coefficient,

`C(n,k) = (n!)/(k!(n-k)!)`

so we have

`C(50,3) + C(50,4) = (50!)/(3!47!) + (50!)/(4!46!) \\\\ = (50!)/(3!46!) \left(\frac1{47} + \frac14\right) \\\\ = (50!)/(3!46!) * (51)/(188) \\\\ = (51!)/(3!46!) * \frac1{4*47} \\\\ = (51!)/(4!47!) \\\\ = (51!)/(4!(51-4)!) = C(51,4)`

as required.