A school has 13 chess players. A team of 4 is chosen, and one of the four is named captain…

For binomial coefficients, ${}^{n}C_{r} = {}^{n}C_{n - r}$, and if ${}^{n}C_{a} = {}^{n}C_{b}$ with $a \ne b$ then $a + b = n$. Pascal's rule is ${}^{n}C_{r - 1} + {}^{n}C_{r} = {}^{n + 1}C_{r}$. Neighbours satisfy $\dfrac{{}^{n}C_{r}}{{}^{n}C_{r - 1}} = \dfrac{n - r + 1}{r}$, and $r \cdot {}^{n}C_{r} = n \cdot {}^{n - 1}C_{r - 1}$.