Calculating binomial coefficients

Statement

Calculate $\dbinom{10}{3}$, $\dbinom{10}{7}$, $\dbinom{10}{0}$, $\dbinom{10}{10}$.

Solution

We use $\binom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$ and the symmetry $\binom{n}{k}=\binom{n}{n-k}$. $\binom{10}{3} = \frac{10\cdot 9\cdot 8}{3!} = \frac{720}{6} = \boxed{120}$ By symmetry $\binom{10}{7} = \binom{10}{3} = \boxed{120}$. Finally, choosing $0$ objects or all $10$ can be done in only one way: $\binom{10}{0} = \boxed{1}, \qquad \binom{10}{10} = \boxed{1}$

Links

Topics: Combinatorics
Concepts: Binomial coefficient · Combinations
Methods: Simple combinations
Skills: Combinatorial calculus · Using formulae
Exercise type: Combinatorial problem