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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
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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
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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
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Mrs. Rivas Evaluate each expression 4. 8! 8!=8∙7∙6∙5∙4∙3∙2∙1=𝟒𝟎,𝟑𝟐𝟎
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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
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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
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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
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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
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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
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Mrs. Rivas Evaluate each expression 10. 12P1
= 12! 12−1 ! = 12! 11! = 12∙11! 11! =𝟏𝟐
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Mrs. Rivas Evaluate each expression 11. 7P4
= 7! 7−4 ! = 7! 3! = 7∙6∙5∙4∙3! 3! =𝟖𝟒𝟎
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Mrs. Rivas Evaluate each expression
= 12! 11! 12−11 ! = 12! 11! 1 ! = 12! 11! = 12∙11! 11! =𝟏𝟐
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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
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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
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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
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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
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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
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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?
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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?
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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?
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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!
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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)!
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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
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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
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Mrs. Rivas 25.Writing Explain the difference between a permutation and a combination.
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