TAOCP 1.2.6 Exercise 42

Section 1.2.6: Binomial Coefficients

Exercise 42. [HM10] Express the binomial coefficient $\binom{r}{k}$ in terms of the beta function.

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By Exercise 40(3),

$$ B(k+1,r-k+1) =\frac{(r+1)!}{k!(r-k)!},B(r+1,1). $$

Since Exercise 40(1) gives $B(r+1,1)=1/(r+1)$, we obtain

$$ B(k+1,r-k+1) =\frac{r!}{k!(r-k)!}. $$

Therefore, by Eq. (5),

$$ \binom{r}{k} =\frac{r!}{k!(r-k)!} =\frac{1}{B(k+1,r-k+1)}, \qquad \text{integer } r \ge k \ge 0. $$

Hence

$$ \boxed{\binom{r}{k}=\frac{1}{B(k+1,r-k+1)}}. $$