1. Mrs. Rivas
1. How many 2-letter pairs of 1 vowel and 1 consonant can you make from the English alphabet? Consider “y” to be a consonant.
There are 26 letter in the English alphabet, 5 vowel and 21 consonants
5  21 = 105

2. Mrs. Rivas
2. An ice cream shop offers 33 flavors of ice cream and 7 toppings. How many different sundaes can the shop make using 1 flavor and 1 topping?
33 flavors and 7 toppings
33  7 = 231

3. Mrs. Rivas
3. A contest winner gets to choose 1 of 8 possible vacations and bring 1 of 10 friends with her. How many different ways could the contest winner select her prize?
8 vacations and 10 friends
8  10 = 80

4. Mrs. Rivas
Evaluate each expression
4. 8!
8!=8∙7∙6∙5∙4∙3∙2∙1=𝟒𝟎,𝟑𝟐𝟎

5. Mrs. Rivas 11! 9! = 11∙10∙9∙8∙7∙6∙5∙4∙3∙2∙1 9∙8∙7∙6∙5∙4∙3∙2∙1 =𝟏𝟏𝟎
Evaluate each expression
5. 11! 9!
11! 9! = 11∙10∙9∙8∙7∙6∙5∙4∙3∙2∙1 9∙8∙7∙6∙5∙4∙3∙2∙1 =𝟏𝟏𝟎

6. Mrs. Rivas 9! 2!6! = 9∙8∙7∙6! 2∙1∙6! = 9∙4∙2∙7 2∙1 =𝟐𝟓𝟐
Evaluate each expression
6. 9! 2!6!
9! 2!6! = 9∙8∙7∙6! 2∙1∙6! = 9∙4∙2∙7 2∙1 =𝟐𝟓𝟐

7. Mrs. Rivas 7. 3!8! 5! 3!8! 5! = 3∙2∙1∙8∙7∙6∙5! 5! =3∙2∙1∙8∙7∙6=𝟐,𝟎𝟏𝟔
Evaluate each expression
7. 3!8! 5!
3!8! 5! = 3∙2∙1∙8∙7∙6∙5! 5! =3∙2∙1∙8∙7∙6=𝟐,𝟎𝟏𝟔

8. Mrs. Rivas Evaluate each expression 8. 12P11 = 12! 12−11 ! = 12! 1!
= 12! 12−11 ! = 12! 1!
=12∙11∙10∙9∙8∙7∙6∙5∙4∙3∙2∙1
=𝟒𝟕𝟗,𝟎𝟎𝟏,𝟔𝟎𝟎

9. Mrs. Rivas Evaluate each expression
= 8! 8−6 ! = 8! 2! = 8∙7∙6∙5∙4∙3∙2! 2!
=8∙7∙6∙5∙4∙3
=𝟐𝟎,𝟏𝟔𝟎

10. Mrs. Rivas Evaluate each expression 10. 12P1
= 12! 12−1 ! = 12! 11! = 12∙11! 11! =𝟏𝟐

11. Mrs. Rivas Evaluate each expression 11. 7P4
= 7! 7−4 ! = 7! 3! = 7∙6∙5∙4∙3! 3! =𝟖𝟒𝟎

12. Mrs. Rivas Evaluate each expression
= 12! 11! 12−11 ! = 12! 11! 1 ! = 12! 11! = 12∙11! 11! =𝟏𝟐

13. Mrs. Rivas Evaluate each expression
= 12! 5! 12−5 ! = 12! 5! 7 ! = 12∙11∙10∙9∙8∙7! 5∙4∙3∙2∙7!
13. 12C5
= 12∙11∙5∙2∙3∙3∙4∙2 5∙4∙3∙2
=𝟕𝟗𝟐

14. Mrs. Rivas Evaluate each expression 14. 5C4 + 5C3
= 5! 4! 5−4 ! = 5! 4! 1 ! = 5∙4! 4! =5
5C4
= 5! 3! 5−3 ! = 5! 3! 2 ! = 5∙4∙3∙2! 3∙2∙1∙2! = 20 2 =10
5C3
5C4 + 5C3 =𝟓+𝟏𝟎=𝟏𝟓

15. Mrs. Rivas Evaluate each expression 15. 4(7C3)
= 7! 2! 7−3 ! = 7! 3! 4 ! = 7∙6∙5∙4! 3∙2∙1∙4! = 7∙6∙5 3∙2 =7∙3=𝟑𝟓
7C3
𝟒(𝟐𝟏)=𝟏𝟒𝟎

16. Mrs. Rivas a) 7∙6∙5∙4∙3∙2∙1=𝟓𝟎𝟒𝟎 b) 6∙5∙4∙3∙2∙1=𝟕𝟐𝟎
16. An art gallery plans to display 7 sculptures in a single row.
a) How many different arrangements of the sculptures are possible?
b) If one sculpture is taken out of the show, how many different arrangements are possible a)
7∙6∙5∙4∙3∙2∙1=𝟓𝟎𝟒𝟎 b)
6∙5∙4∙3∙2∙1=𝟕𝟐𝟎

17. Mrs. Rivas
17. Thirty people apply for 10 job openings as welders. How many different groups of people can be hired?
= 30! 10! 30−10 ! = 30! 10! 20 !
30C10 𝟑 𝟕 𝟑 𝟓 𝟑 𝟏𝟏 𝟑
= 30∙29∙28∙27∙26∙25∙24∙23∙22∙21∙20! 10∙9∙8∙7∙6∙5∙4∙3∙2∙1∙20!
= 3∙29∙7∙3∙26∙5∙3∙23∙11∙3 6∙3 = 18,027,009 6
=𝟑𝟎,𝟎𝟒𝟓,𝟎𝟏𝟓

18. Mrs. Rivas
For each situation, determine whether to use a permutation or a combination. Then solve the problem
18. You draw the names of 5 raffle winners from a basket of 50 names. Each person wins the same prize. How many different groups of winners could you draw?

19. Mrs. Rivas
For each situation, determine whether to use a permutation or a combination. Then solve the problem
19. A paint store offers 15 different shades of blue. How many different ways could you purchase 3 shades of blue?

20. Mrs. Rivas
For each situation, determine whether to use a permutation or a combination. Then solve the problem
20. How many different 5-letter codes can you make from the letters in the word cipher?

21. Mrs. Rivas
Assume a and b are positive integers. Determine whether each statement is true or false. If it is true, explain why. If it is false, give a counterexample.
21. a!b!  b!a!

22. Mrs. Rivas
Assume a and b are positive integers. Determine whether each statement is true or false. If it is true, explain why. If it is false, give a counterexample.
22. a • b!  (ab)!

23. Mrs. Rivas
Assume a and b are positive integers. Determine whether each statement is true or false. If it is true, explain why. If it is false, give a counterexample.
23. (a2)!  (a!)2

24. 5 28 3 = 4 Mrs. Rivas 𝟏𝟔𝟖𝟎    appetizer entrée sides desserts  4C1
24. A restaurant offers a fixed-priced meal of 1 appetizer, 1 entrée, 2 sides, and 1 dessert. How many different meals could you choose from 4 appetizers, 5 entrees, 8 sides, and 3 desserts?   
appetizer
entrée
sides
desserts 
4C1
5C1 
8C2 
3C1 5 28 3 = 4   
𝟏𝟔𝟖𝟎

25. Mrs. Rivas
25.Writing Explain the difference between a permutation and a combination.