Choosing Team Members

A club has 5 members. How many ways can you choose a 2-person committee?

Identify n and r: $n = 5$ (total members), $r = 2$ (to choose) = $n = 5$, $r = 2$

Apply the combination formula: $C(5,2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \times 3!}$ = Formula set up

Calculate factorials: $\frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{5 \times 4}{2}$ = $\frac{20}{2}$

Simplify: $\frac{20}{2} = 10$ = $10$ ways

Answer: There are 10 different 2-person committees possible.

Pizza Toppings

A pizza shop offers 8 toppings. How many different 3-topping pizzas can you make?

Identify n and r: $n = 8$ (total toppings), $r = 3$ (to choose) = $n = 8$, $r = 3$

Apply the formula: $C(8,3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \times 5!}$ = Formula set up

Simplify factorials: $\frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1}$ = $\frac{336}{6}$

Calculate: $336 \div 6 = 56$ = $56$ pizzas

Answer: You can create 56 different 3-topping pizzas.

Lottery Selection

In a lottery, 6 numbers are drawn from 49. How many possible outcomes are there?

Identify n and r: $n = 49$ (total numbers), $r = 6$ (drawn) = $n = 49$, $r = 6$

Set up the formula: $C(49,6) = \frac{49!}{6! \times 43!}$ = Formula ready

Expand numerator: $\frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6!}$ = $\frac{10,068,347,520}{720}$

Calculate: $10,068,347,520 \div 720 = 13,983,816$ = $13,983,816$ outcomes

Answer: There are 13,983,816 possible lottery outcomes, which is why winning is so rare!

Mistake: Confusing combinations with permutations

Why: Permutations count arrangements (order matters), combinations count selections (order doesn't matter). $P(5,2) = 20$ but $C(5,2) = 10$.

Correct: Ask yourself: 'Does rearranging change the outcome?' If choosing a committee, {A,B} = {B,A}, so use combinations.

Mistake: Forgetting to divide by $r!$ in the formula

Why: The division by $r!$ removes the duplicate orderings of the same selection.

Correct: Always use the full formula: $C(n,r) = \frac{n!}{r!(n-r)!}$

Mistake: Calculating factorials incorrectly

Why: Large factorials can be overwhelming. Remember that most terms cancel out.

Correct: Simplify before multiplying: $\frac{7!}{5! \times 2!} = \frac{7 \times 6}{2} = 21$

Sports Team Selection

Coaches use combinations to understand how many different team lineups are possible.

A basketball coach choosing 5 starters from 12 players: $C(12,5) = 792$ possible lineups.

Card Games

Combinations calculate the number of possible hands in card games.

A 5-card poker hand from 52 cards: $C(52,5) = 2,598,960$ possible hands.

Menu Planning

Restaurants use combinations to calculate meal options and combo deals.

Choosing 3 appetizers from 10 options: $C(10,3) = 120$ different selections.

Q: When do I use combinations vs permutations?

A: Use combinations when order doesn't matter (selecting a committee, choosing toppings). Use permutations when order matters (assigning positions, arranging a lineup in order).

Q: Why is $C(n,r) = C(n, n-r)$?

A: Choosing $r$ items to include is the same as choosing $n-r$ items to exclude. For example, $C(5,2) = C(5,3) = 10$.

Q: What does the '!' mean in the formula?

A: The exclamation mark means factorial. $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. It counts all possible arrangements of a set.