Combinations

Definition

$C_{r}^{n} = \frac{n!}{r! (n - r)!}$

Order doesn't matter, we don't distinguish between objects.
Example:
- Calculate the number of ways to select $3$ apples from a basket of $5$ apples
- Non distinct for this example
- $\begin{matrix} A p p l e s & 1 & 2 & 3 & 4 & 5 \\ ✓ & ✓ & ✓ \\ ✓ & ✓ & ✓ \\ ✓ & ✓ & ✓ \end{matrix}$
- $C_{3}^{5} = \frac{5!}{3! (5 - 3)!} = \frac{120}{6 (2)} = 10$
Expansion
- If you have $(a_{1} + a_{2})^{2} = a_{1}^{2} + 2 a_{1} a_{2} + a_{2}^{2}$
- $(a_{1} + a_{2})^{3} = a_{1}^{3} + \frac{3!}{1! 1!} a_{1}^{2} a_{2} \dots$
- Multinomial Theorem.