Math: Permutations and Combinations
Counting outcomes with order and selection
Math: Permutations and Combinations
Counting outcomes with order and selection
Math - Grade 9-12
- 1
How many different ways can 5 books be arranged on a shelf?
When order matters for all items, use a factorial.
There are 120 different arrangements because 5! = 5 x 4 x 3 x 2 x 1 = 120. - 2
A school elects a president, vice president, and secretary from 8 students. How many different ways can the officers be chosen?
There are 336 ways to choose the officers because order matters, so the count is 8P3 = 8 x 7 x 6 = 336. - 3
How many ways can a committee of 4 students be chosen from a group of 10 students?
A committee is a selection, not an arrangement.
There are 210 ways to choose the committee because order does not matter, so the count is 10C4 = 210. - 4
How many 3-letter arrangements can be made from the letters A, B, C, D, and E if no letter is repeated?
There are 60 arrangements because order matters and no repetition is allowed, so 5P3 = 5 x 4 x 3 = 60. - 5
How many different 2-topping pizzas can be made from 6 available toppings if no topping is used more than once?
The order of the toppings does not change the pizza.
There are 15 different pizzas because choosing toppings is a combination, so 6C2 = 15. - 6
A 4-digit code is made using the digits 1 through 9 with no repeated digits. How many codes are possible?
There are 3024 possible codes because order matters and no digits repeat, so 9P4 = 9 x 8 x 7 x 6 = 3024. - 7
How many ways can 7 runners finish in first, second, and third place?
Award places are ordered positions.
There are 210 ways because the order of first, second, and third matters, so 7P3 = 7 x 6 x 5 = 210. - 8
From 12 songs, how many different playlists of 5 songs can be chosen if the order does not matter?
There are 792 playlists because this is a selection of 5 songs from 12, so 12C5 = 792. - 9
How many ways can the letters in the word MATH be arranged?
Count the number of ways to arrange 4 distinct letters.
There are 24 arrangements because all 4 letters are different, so 4! = 24. - 10
A class has 14 students. In how many ways can 2 students be chosen to represent the class at a meeting?
There are 91 ways to choose the representatives because order does not matter, so 14C2 = 91. - 11
How many 5-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5 if no digit is repeated and the first digit cannot be 0?
Choose the first digit carefully before arranging the rest.
There are 600 such numbers. There are 5 choices for the first digit because it cannot be 0, then 5, 4, 3, and 2 choices for the remaining places, so 5 x 5 x 4 x 3 x 2 = 600. - 12
How many ways can 3 science books and 2 math books be arranged on a shelf if all 5 books are different?
There are 120 arrangements because all 5 books are different, so the number of ways is 5! = 120. - 13
A restaurant menu has 9 entrees. How many different groups of 4 entrees can be chosen for a special tasting if no entree is repeated?
A group for tasting is a combination.
There are 126 groups because order does not matter, so 9C4 = 126. - 14
How many ways can 6 students stand in a line for a photo?
There are 720 ways because order matters in a line, so 6! = 720. - 15
A bag contains 8 different colored marbles. How many ways can 3 marbles be selected if the order of selection does not matter?
Selection without order uses combinations.
There are 56 ways to select the marbles because this is a combination, so 8C3 = 56.