# Math 30-1: Permutations and Combinations PRACTICE EXAM

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1 A. B. C. D. Math 30-1: Permutations and Combinations PRACTICE EXAM n! 1. The expression is equivalent to: (n - 2)! 2. A Grade 12 student is taking Biology, English, Math, and Physics in her first term. If a student timetable has room for five courses (meaning the student has a spare), how many ways can she schedule her courses? A. 24 B. 120 C. 240 D. 720 One Possible Timetable Block Block 1 Block 2 Block 3 Block 4 Block 5 Course Math 30-1 Spare Physics 30 English 30-1 Biology An electrical panel has five switches. How many ways can the switches be positioned up or down if three switches must be up and two must be down? One possible switch arrangement. A. 10 B. 24 C. 48 D A coat hanger has four knobs, and each knob can be painted any color. If six different colors of paint are available, how many ways can the knobs be painted? A. 24 B. 360 C. 720 D. 1296

2 5. How many ways can the letters from the word TREES be ordered such that each word starts with a consonant and end with a vowel? A. 9 B. 18 C. 24 D How many arrangements of the word ACTIVE are there if C&E must always be together? A. 48 B. 120 C. 240 D Eight cars (3 red, 3 blue, and 2 yellow) are to be parked in a line. How many unique lines can be formed if the yellow cars must not be together? Assume that cars of each color are identical. A. 18 B. 420 C. 560 D How many 3-digit odd numbers greater than 600 can be formed using the digits (2, 3, 4, 5, 6, and 7)? A. 20 B. 36 C. 120 D. 720 n! 9. The equation has the solution: 10 = P n-1 n-3 A. n = 5 B. n = 6 C. n = 7 D. n = 8

3 10. There are 9 dots randomly placed on a circle. How many triangles can be formed within the circle? A. 84 B. 120 C. 720 D A crate of toy cars contains 10 working cars and 4 defective cars. How many ways can 5 cars be selected if only 3 work? A. 6 B. 56 C. 720 D A committee of 5 people is to be formed from a selection pool of 12 people. If Carmen must be on the committee, how many unique committees can be formed? A. 60 B. 330 C. 462 D How many five-letter words using letters from TRIANGLE can be made if the five-letter word must have two vowels and three consonants? A. 56 B C D Twelve people at a party shake hands once with everyone else in the room. How many handshakes took place? A. 66 B. 132 C. 12! 2 D. 12!

4 15. A jar contains quarters, loonies, and toonies. If four coins are selected from the jar, how many unique coin combinations are there? A. 15 B. 18 C. 21 D From a deck of 52 cards, a 5-card hand is dealt. How many distinct hands can be formed if there are at most 2 queens? A B C D In how many ways can you choose one or more of 5 different candies? A. 16 B. 25 C. 31 D The solution to is: A. n = 2 B. n = 3 C. n = 4 D. n = The solution to = 1 is: A. n = 2 B. n = 3 C. n = 4 D. n = 5 n C 4 n - 2 C 2

5 20. If there are three cars and four motorcycles, how many ways can the vehicles park in a line such that cars and motorcycles alternate positions? A. 35 B. 70 C. 144 D There are nine people participating in a raffle. Three \$50 gift cards from the same store are to be given out as prizes. How many ways can the gift cards be awarded? A. 84 B. 504 C. 720 D A set of tiles contains eight letters, A - H. If two of these sets are combined, how many ways can all the tiles be arranged? A. B. C. D. 23. Moving only south and east, how many unique pathways connect points A and C? A A. 9 B. 36 C. 84 D. 120 B C

6 24. A bookshelf has n fiction books and six non-fiction books. If there are 150 ways to choose two books of each type, how many fiction books are on the bookshelf? A. 5 B. 6 C. 7 D A particular college in Alberta has a list of approved pre-requisite courses: Math Science English Other Math 30-1 or Math 30-2 Biology 30 Chemistry 30 Physics 30 English 30-1 Option A Option B Option C Option D Option E Five courses are required for admission to the college. Math 30-1 (or Math 30-2) and English 30-1 are mandatory requirements, and at least one science course must be selected as well. How many different ways could a student select five courses on their college application form? A. 88 B. 90 C. 92 D If a 5-card hand is dealt from a deck of 52 cards, how many hands have cards that are all the same color? A B C D A multiple choice test contains 5 questions, and each question has four possible responses. How many different answer keys are possible? A. 20 B. 120 C. 256 D. 1024

7 28. The number of diagonals in a pentagon is: A. 5 B. 6 C. 7 D How many ways can eight books, each covering a different subject, be arranged on a shelf such that books on biology, history, or programming are never together? A B C D Five different types of fruit and six different types of vegetables are available for a healthy snack tray. The snack tray is to contain two fruits and three vegetables. How many different snack trays can be made if blueberries or carrots must be served, but not both together? A. 100 B. 120 C. 140 D In genetics, a codon is a sequence of three letters that specifies a particular amino acid. A fragment of a particular protein yields the amino acid sequence: Met - Gly - Ser - Arg - Cys - Gly. How many unique codon arrangements could yield this amino acid sequence? A B C D Amino Acid Arginine (Arg) Cysteine (Cys) Glycine (Gly) Methionine (Met) Serine (Ser) Codon(s) CGU, CGC, CGA, CGG, AGA, AGG UGU, UGC GGU, GGC, GGA, GGG AUG UCU, UCC, UCA, UCG, AGU, AGC

8 32. The third term in row 22 of Pascal s Triangle is: A. 21 C 2 B. 21 C 3 C. 22 C 2 D. 22 C Which of the following is not a term in the expansion of (2x - 3) 4? A. 16x 4 B. -96x 3 C. -216x 2 D. -216x 34. Which of the following is not a term in the expansion of? A. B. C. D. 35. The expression 32a 5-240a 4 b + 720a 3 b a 2 b ab 4-243b 5 is equivalent to: A. (2a - 3b) 3 B. (2a - 3b) 4 C. (2a - 3b) 5 D. (2a - 3b) The fifth term in the expansion of (3a 3-2b 2 ) 8 is: A a 12 b 8 B a 12 b 8 C a 9 b 10 D a 9 b 10

9 37. In the expansion of (5a - 2b) 9, the coefficient of the term containing a 5 is: A B C D If there are 23 terms are in the expansion of (a - 2) 3k-5, the value of k is: A. 8 B. 9 C. 10 D A term in the expansion of (ma - 4) 5 is -5760a 2. What is the value of m? A. -3 B. 3 C. -5 D In the expansion of, the constant term is: A. -30 B. 30 C. -60 D. 60

11 - ANSWER KEY Video solutions are in italics. 1. A Permutations, Example 2g 2. B Permutations, Example 3a 3. A Permutations, Example 4f 4. D Permutations, Example 5c 5. B Permutations, Example 7b 6. C Permutations, Example 8b 7. B Permutations, Example 9b 8. A Permutations, Example 10c 9. A Permutations, Example 12c 10. A Combinations, Example 2d (ii) 11. C Combinations, Example 3b 12. B Combinations, Example 4a 13. C Combinations, Example 5a 14. A Combinations, Example 6a 15. A Combinations, Example 7a 16. C Combinations, Example 8b 17. C Combinations, Example 8e 18. D Combinations, Example 10c 19. C Combinations, Example 11a 20. C Combinations, Example 12d 21. A Combinations, Example 12f 22. D Combinations, Example 13a 23. D Combinations, Example 13e 24. A Combinations, Example 14b 25. C Combinations, Example 14e 26. C Combinations, Example 15c 27. D Combinations, Example 15e 28. A Combinations, Example 15f 29. B Combinations, Example 15g 30. A Combinations, Example 16c 31. B Combinations, Example 16d 32. A The Binomial Theorem, Example 2b 33. C The Binomial Theorem, Example 4c 34. C The Binomial Theorem, Example 5b 35. C The Binomial Theorem, Example 6b 36. B The Binomial Theorem, Example 7b 37. D The Binomial Theorem, Example 8a 38. B The Binomial Theorem, Example 8d 39. B The Binomial Theorem, Example 9a 40. D The Binomial Theorem, Example 10a

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