Combinations

For each of the problems below, (1) figure out whether this involves combination with or without replacement, then (2) compute the solution.

	Word problem	Replacement?	Solution
1	The shaved ice truck has added six new flavors, and three friends want to sample them. They agree to order different flavors, and then all three will try each flavor. How many possible combinations are there?	Yes No	$\displaystyle combinations(6, 3) = \frac{6!}{(6-3)!} \div 3$ $\displaystyle = \frac{6!}{3!} \div 3 = \frac{720}{6} \div 3$ $\displaystyle = 120 \div 3 = 40$
2	A soccer team has 20 players, but only 11 are allowed on the field at once. How many different groups of players can be on the field at one time?	Yes No
3	A set of pool balls is numbered 1-15. How many different ways are there to choose six balls?	Yes No
4	Six friends get together to play video games. All the games can only have two players, so they decided to pair off to make sure everyone gets to play everyone else. How many games will they have to play?	Yes No
5	A set of pool balls is numbered 1-15. Seven of them are striped and eight are solid colors. How many different ways are there to choose 4 balls where 2 are striped and 2 are solid?	Yes No
★	A pizzeria has a 3-topping special on any pizza, for only $12.99. If they have 10 toppings to choose from, how many different pizzas can they make?	Yes No

Word problem

Replacement?

Solution

The shaved ice truck has added six new flavors, and three friends want to sample them. They agree to order different flavors, and then all three will try each flavor. How many possible combinations are there?

Yes No

$\displaystyle combinations(6, 3) = \frac{6!}{(6-3)!} \div 3$ $\displaystyle = \frac{6!}{3!} \div 3 = \frac{720}{6} \div 3$
$\displaystyle = 120 \div 3 = 40$

A soccer team has 20 players, but only 11 are allowed on the field at once. How many different groups of players can be on the field at one time?

Yes No

A set of pool balls is numbered 1-15. How many different ways are there to choose six balls?

Yes No

Six friends get together to play video games. All the games can only have two players, so they decided to pair off to make sure everyone gets to play everyone else. How many games will they have to play?

Yes No

A set of pool balls is numbered 1-15. Seven of them are striped and eight are solid colors. How many different ways are there to choose 4 balls where 2 are striped and 2 are solid?

Yes No

★

A pizzeria has a 3-topping special on any pizza, for only $12.99. If they have 10 toppings to choose from, how many different pizzas can they make?

Yes No

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