Algebra 2 / Unit 9: Probability
9.3 Combinations
Notes
A combination is a selection where order does not matter.
- $\displaystyle _nC_r = \frac{n!}{r!\,(n - r)!}$.
- It equals $_nP_r$ divided by $r!$, since each group can be ordered $r!$ ways.
- Ask: "If I swap two chosen items, is it a different outcome?" Yes means permutation; no means combination.
- Example: choosing 2 of 5 friends to invite: $\displaystyle _5C_2 = \frac{5 \cdot 4}{2} = 10$.
Practice
Evaluate $_{10}C_3$.
$\displaystyle \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120$.
How many 4-person committees can be chosen from 9 people?
$\displaystyle _9C_4 = \frac{9 \cdot 8 \cdot 7 \cdot 6}{24} = 126$.
Which situation is counted with a combination?
The order of the toppings doesn't change the pizza.
Ready for more? Take the Unit 9 test.