Permutations, Combinations And Binomial Theorem Exam Questions
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1 Permutations, Combinations And Binomial Theorem Exam Questions Name: ANSWERS
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3 Multiple Choice 1. Find the total possible arrangements for 7 adults and 3 children seated in a row if the 3 children must be together. a. 10! b. 8!3! c. 7!3! d. 7! 7. How many terms are in the expansion of 3y 4z? a. b. 6 c. 7 d How many different arrangements are possible when arranging all the letters of the word SEPTEMBER? 9! a. 9! b. 6!3! c. 3! d. 6! 3! 4. There are 13 terms in the expansion of 3x y. Determine the values of n. n a. 6 b. 6.5 c. 7 d. 6 3
4 10 5. How many terms are there in the expansion of x 1 3? a. 9 b. 10 c. 11 d If C, the value of n must be: n 5 n C 3 a. 3 b. 5 c. 8 d There are types of pencils, 3 colours of highlighters, and 5 styles of pens. If you must select one of each to form a set, how many different sets of writing instruments are possible? a. 10 b. 11 c. 5 d. 30 4
5 8. Given the following row of Pascal s Triangle, identify the binomial expansion with these coefficients a. x y b. x y c. x y d. x y Identify which of the following represents the 5 th 3 term in the expansion of 4x y a. C 4x y b x y 4 15C c. C 4x y d x y 4 15C4 10. There are 6 different books that are being distributed evenly amongst three people. Identify which expression represents the number of possible combinations. C C C 6C 4C C a. b C C C 3 6C c. d. 5
6 Written Response 11. A school offers 4 different Science courses, 3 different Mathematics courses, and different English courses. Julie must select 1 Science course, 1 Mathematics course, and 1 English course. She thinks this creates 9 options for her timetable. Show why Julie is incorrect. (1 mark) 1. Explain how Pascal s triangle can be used to determine the coefficients in the binomial expansion of x y n. (1 mark) 13. Bella has pairs of shoes, 3 pairs of pants, and 10 shirts. Carey has 4 pair of shoes, 4 pairs of pants, and 4 shirts. An out it is made up of one pair of shoes, one pair of pants, and one shirt. Who can make more outfits? Justify your answer. (1 mark) 6
7 In the binomial expansion of x y, how many terms will be positive? Justify your answer. ( mark) Evaluate the coefficient of the term containing x in the expansion of 1 x. Justify your answer. ( marks) Find and simplify the last term in the expansion of y 3x. ( marks) (calculator) 7
8 17. How many different ways can 4 girls and 4 boys be arranged in a row if the girls and the boys must alternate? (calculator) ( marks) 18. A particular math class has a large number of students. From this class, you are to create a committee of 4 students that has at least 1 girl. Without actually solving the problem, explain the strategy you would use to find the total number of ways to select this committee. ( marks) 8
9 19. There are 5 men and 4 women to be seated in a row. How many arrangements are possible if two men must sit at the beginning of the row and two men must sit at the end of the row? (calculator) ( marks) 0. Explain why a locker combination should really be called a locker permutation. (1 mark) 1. You are given the following row of Pascal s Triangle Determine the values of the next row. (1 mark) 9
10 . Solve algebraically: (3 marks) 4n 5 C n 3. A word contains two Ms, two Es, and two Ns, and no other repeated letters. Suppose one of the Ns is replaced by an M. Will this replacement result in greater or fewer permutations? Justify your reasoning. (calculator) ( marks) 10
11 4. There is a group of 16 boys and 1 girls. How many ways can a committee of 3 people be formed if there must be at least girls on the committee? (calculator) (3 marks) 5. Find and simplify the 6 th term in the binomial expansion of 3x 3. (3 marks) (calculator) x
12 6. The 4 th 3 term in the binomial expansion of is. qx x x 11 Determine the value of q algebraically. (calculator) 10 (3 marks) 7. Solve the following equation: (3 marks) P n n C 3 1
13 a. In the binomial expansion of 4x, determine the 3 rd term. (3 marks) (calculator) x 8 n b. In the binomial expansion of 4x, the 6 th term contains. ( marks) x Solve for n. (calculator) x 13
14 9. Solve: (3 marks) 4 P n1 30. a. From a group of 9 people, in how many ways can you select a committee of 4 members? (calculator) (1 mark) b. From a group of 9 people, in how many ways can you select a president, a vice president, a secretary, and a treasurer? (1 mark) (calculator) c. Explain why the answers in a and b are different. (1 mark) 14
15 31. Simplify the 6 th term in the expansion of: (calculator) (3 marks) 3 x x 10 7 P 3. Evaluate: P (1 mark)
16 33. In how many different ways can you arrange the letters in the word VOLLEYBALL? State your answer as a factorial. (1 mark) 34. Determine the number of possible sandwiches from the following menu. (1 mark) 16
17 35. David and Sarah are in a class of 10 boys and 8 girls. A committee of 3 boys and girls is to be selected from the students in this class. Determine the number of possible committees if David and Sarah cannot be on the same committee. (calculator) (3 marks) In the binomial expansion of x, simplify the 7 th term. (calculator) x 10 (3 marks) 17
18 37. Explain why is undefined. (1 mark) 3 C Solve: n P3 48 n 1 (3 marks) 18
19 39. There are 9 girls and 7 boys in a math class from which a committee of 5 is to be chosen. a. How many different committees of 5 can be formed if one of the boys, William, must be on the committee? (calculator) (1 mark) Solution: C C b. How many different committees of 5 can be formed if there must be girls and 3 boys on the committee? (calculator) ( marks) Solution: 9C7 C A hockey arena has 5 doors. Determine the number of ways that you can enter through one door but exit through a different door. (1 mark) Solution: 5 4 = 0 19
20 In the binomial expansion of x 3, determine which term contains x. x Solution: the 7 th term contains 3 x 9 (you must show work) ( marks) 4. Solve algebraically: n P3 4! n 1 (3 marks) Solution: n! 4! n 1 n 3! nn 1n n 3! 4! n 3! nn 1n 4! n 1 nn 4 n n 4 0 n 6n 4 0 n 6 n 4 no possible n 1 0
21 8 43. Determine and simplify the fourth term in the expansion of 4 x 3y. (3 marks) (calculator) 44. An employee asked 10 people in an ice cream shop to wait in line. Determine the number of different arrangements possible if two of the people, Jamie and John, refuse to stand next to each other in the line. ( marks) 1
22 45. Raoul has 8 shirts, 5 pairs of pants, and 3 hats. He adds the options together and determines that he has 16 different outfits to wear. Raoul made an error in calculating the number of different outfits. Describe how to determine the correct number of outfits. (1 mark) Justify why the binomial expansion of x x does not have a term that contains x. ( marks)
23 47. Solve algebraically: C n n 3 (3 marks) 3
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