Factorials, Permutations and Combinations

Problem: 1

Evaluate the expression 6!.

Problem: 3

Find the value of 15!

Problem: 5

Simplify the following:

7! 7

Problem: 7

Evaluate:

Problem: 9

Simplify.

(k + 3)!(k + 4)

Problem: 11

Simplify.

Problem: 13

Determine whether the given statement is true or false, and give a counterexample if the statement is false:

Problem: 15

If x and y are positive integers determine whether the statement x! – y! = y! – x! is true or false. If the statement is true, explain. If the statement is false, give a counterexample.

Problem: 17

Determine whether the given statement is true or false, and give a counterexample if the statement is false:

(5 x)! = 5^x(x!)

Problem: 19

Evaluate the expression ₉P₁.

Problem: 21

Evaluate the expression ₇P₄.

Problem: 23

Simplify.

₈C₄

Problem: 25

Simplify.

₁₂C₉

Problem: 27

Which is greater?

Problem: 29

Solve:

Problem: 31

A group of students standing in a row. State whether this is a permutation or a combination.

Problem: 33

Rajesh ordered five items from a menu in a restaurant. State whether this is a permutation or a combination.

Problem: 35

Obtain the number of permutations possible for the group of letters shown. Use all the letters.

F, I, V, E

Problem: 37

Obtain the number of three–letter permutations possible for the group of letters shown.

S, E, V, E, N

Problem: 39

Find the number of combinations using two different letters that can be formed from the group of letters by listing the combinations.

F, O, U, R

Problem: 41

In how many different ways can the letters of the word EXPRESSION be arranged?

Problem: 43

How many number of 6–letter permutations be formed from the letters in the word FLUBBER?

Problem: 45

A choir of 10 students have to be chosen from a class of 17. State whether this is a permutation or a combination and find the number of ways it can be done.

Problem: 47

Check whether the given arrangement is linear or circular. Check if the given arrangement is also a reflection and find the number of arrangements.

A basketball huddle of 7 players

Problem: 49

Check whether the given arrangement is linear or circular. Check if the given arrangement is also a reflection and find the number of arrangements.

9 pizza toppings placed on a revolving tray.

Problem: 51

Stephen has to pick 5 fruits from the set of fruits

{apple, banana, mango, orange, peach, strawberry).

Find the number of combinations.

Problem: 53

5 students from 100 participants have to be chosen for the International Mathematics Olympiad team. Find the number of combinations.

Problem: 55

How many 13–card hands with exactly 11 spades can be dealt?

Problem: 57

How many 6–card hands having exactly 3 aces and 3 other cards can be dealt? (Assume you have a regular 52–card deck.)

Problem: 59

Find in how many ways can one select 5 or more students from 9 students?

Problem: 61

Get the number of possible 6–card hands that contain the cards 4 jacks and 2 other cards.

Problem: 1

Evaluate the expression 6!.

Problem: 3

Find the value of 15!

Problem: 5

Simplify the following:

7! 7

Problem: 7

Evaluate:

Problem: 9

Simplify.

(k + 3)!(k + 4)

Problem: 11

Simplify.

Problem: 13

Determine whether the given statement is true or false, and give a counterexample if the statement is false:

Problem: 15

If x and y are positive integers determine whether the statement x! – y! = y! – x! is true or false. If the statement is true, explain. If the statement is false, give a counterexample.

Problem: 17

Determine whether the given statement is true or false, and give a counterexample if the statement is false:

(5x)! = 5^x(x!)

Problem: 19

Evaluate the expression ₉P₁.

Problem: 21

Evaluate the expression ₇P₄.

Problem: 23

Simplify.

₈C₄

Problem: 25

Simplify.

₁₂C₉

Problem: 27

Which is greater?

Problem: 29

Solve:

Problem: 31

A group of students standing in a row. State whether this is a permutation or a combination.

Problem: 33

Rajesh ordered five items from a menu in a restaurant. State whether this is a permutation or a combination.

Problem: 35

Obtain the number of permutations possible for the group of letters shown. Use all the letters.

F, I, V, E

Problem: 37

Obtain the number of three–letter permutations possible for the group of letters shown.

S, E, V, E, N

Problem: 39

Find the number of combinations using two different letters that can be formed from the group of letters by listing the combinations.

F, O, U, R

Problem: 41

In how many different ways can the letters of the word EXPRESSION be arranged?

Problem: 43

How many number of 6–letter permutations be formed from the letters in the word FLUBBER?

Problem: 45

A choir of 10 students have to be chosen from a class of 17. State whether this is a permutation or a combination and find the number of ways it can be done.

Problem: 47

Check whether the given arrangement is linear or circular. Check if the given arrangement is also a reflection and find the number of arrangements.

A basketball huddle of 7 players

Problem: 49

Check whether the given arrangement is linear or circular. Check if the given arrangement is also a reflection and find the number of arrangements.

9 pizza toppings placed on a revolving tray.

Problem: 51

Stephen has to pick 5 fruits from the set of fruits

{apple, banana, mango, orange, peach, strawberry).

Find the number of combinations.

Problem: 53

5 students from 100 participants have to be chosen for the International Mathematics Olympiad team. Find the number of combinations.

Problem: 55

How many 13–card hands with exactly 11 spades can be dealt?

Problem: 57

How many 6–card hands having exactly 3 aces and 3 other cards can be dealt? (Assume you have a regular 52–card deck.)

Problem: 59

Find in how many ways can one select 5 or more students from 9 students?

Problem: 61

Get the number of possible 6–card hands that contain the cards 4 jacks and 2 other cards.

Hotmath Practice Problems