Word Problems
(Probability, Permutations & Combinations)
Practice Exercise
Permutations & Combinations:
Very Easy (Difficulty Level-1):
1. Amy and Adam are making boxes of truffles to give out as wedding favors. They have an unlimited supply of 5 different types of truffles. If each box holds 2 truffles of different types, how many different boxes can they make ?
2. In how many ways can the letters in the word “Mission” be arranged?
Easy (Difficulty Level-2):
3. A pod of 6 dolphins always swims single file, with 3 females at the front and 3 males in the rear. In how many different arrangements can the dolphins swim?
A. 20
B. 36
C. 40
D. 18
E. 54
4. A delegation from Gotham City goes to Metropolis to discuss a limited Batman-Superman partnership. If the mayor of Metropolis chooses 3 members of the 7-person delegation to meet with Superman, how many different 3-person combinations can he choose?
5. The New York Philharmonic is designing the liner notes for an upcoming CD release. There are 10 soloists featured on the album, but the liner notes are only 5 pages long, and therefore only have room for 5 of the soloists. The soloists are fighting over which of them will appear in the liner notes, as well as who will be featured on which page. How many different liner note arrangements are possible?
Easy (Difficulty Level-3):
6. A men’s basketball league assigns every player a two-digit number for the back of his jersey. If the league uses only the digits 1-5, what is the maximum number of players that can join the league such that no player has a number with a repeated digit (e.g. 22), and no two players have the same number?
(A) 21
(B) 20
(C) 22
(D) 24
(E) 25
7. The Natural Woman, a women’s health food store, offers its own blends of trail mix. If the store uses 4 different ingredients, how many bins will it need to hold every possible blend, assuming that each blend must have at least two ingredients? (Also assume that each bin can hold one and only one blend.)
(A) 1
(B) 4
(C) 7
(D) 11
(E) 10
8. A British spy is trying to escape from the prison cell. The lock code requires him to enter one number, from 1 to 9, and then push a pair of colored buttons simultaneously. There are a total of 6 colored buttons. He can make 1 attempt every 3 seconds. What is the longest possible time it may take him to escape from the cell.
Hard (Difficulty Level-4):
9. Mario’s Pizza has two choices of crust: deep dish and thin-and-crispy. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. Finally, Mario’s offers very pizza in extra-cheese as well as regular. If Linda’s volleyball team decides to order a pizza with four toppings, how many different choices do the teammates have at Mario’s Pizza?
10. Every morning, Casey walks from her house to the bus stop. She always travels exactly nine blocks from her house to the bus, but she varies the route she takes every day. (One sample route is shown). How many days can Casey walk from her house to the bus stop without repeating the same route? (See figure below)
11. 3 dwarves and 3 Elves sit down in a row of 6 chairs. If no dwarf will sit next to another dwarf and no elf will sit next to another elf, in how many different ways can the elves and dwarves sit?
Very Hard & Tricky (Difficulty Level-5):
12. The principal of a high school needs to schedule observations of 6 teachers. she plans to visit one teacher each day for a work week (M-F) so will only have time to see 5 of the teachers. How many different observation schedules can she create?
13. A second grade class is writing reports on birds. The students’ teacher has given them a list of 6 birds they can choose to write about. If Lizzie wants to write a report that includes two or three of the birds, how many different reports can she write?
14. A mother bird has 6 babies.Every time she returns to the nest,she feeds half the babies.If it takes her 5 mins to feed each baby,how long will it take her to feed all the babies?
15. Gordon buys 5 dolls for his 5 nieces. The gifts include two identical S beach dolls, one E, one G, one T doll. If the youngest niece doesn’t want the G doll, in how many different ways can he give the gifts?
Probabilities:
Very Easy (Difficulty Level-1):
16. What is the probability that the sum of two dice will yield a 4 or 6?
17. A jar contains 10 blue marbles, 8 green marbles, and 14 red marbles. How many green marbles must be added so that the probability of choosing a green marble at random is 3⁄4?
(A) 16
(B) 32
(C) 40
(D) 64
(E) 72
18. The probability of a meteor shower occurring in the skies above a particular island on any given night is 2⁄25. Independently, the probability that any given night will be cloudless is 1⁄4. What is the probability that, on any given night, there will be a meteor shower and it will be cloudless?
(A) 1⁄50
(B) 3⁄100
(C) 17⁄200
(D) 4⁄25
(E) 8⁄25
Easy (Difficulty Level-2):
19. What is the probability that the sum of two dice, after thrown, gives a 10 or lower?
20. What is the probability that sum of two dice will yield a 7, and then when both are thrown again, their sum will again yield a 7?
21.At a certain pizza, 1⁄6 of the pizzas sold in a week were cheese, and 1⁄5 of the other pizzas sold were pepperoni. If Zach bought a pizza from the pizzeria that week, what is the probability that he ordered a pepperoni?
Medium (Difficulty Level-3):
22. A six sided die has faces bearing the numbers 1, 2, 3, 4, 5, and 6. When the die is thrown, the numbers on the five visible faces are added. What is the probability that this sum is greater than 18?
23. A basket contains red, green, and yellow balls, all of equal size. The probability of choosing a green ball at random is 4⁄7. If there are three times as many red balls as yellow balls, what is the probability of choosing a yellow ball at random?
24. When a diver dives, there is a 20% chance of a perfect score. However, if a perfect score is given out, the judges won’t give out another. Janet is the third diver to dive. What is the chance of Janet receiving a perfect score?
25. In a bag of marbles, there are 3 red, 2 white, and 5 blue. If Bob takes 2 marbles out of the bag, what is the probability that he will have one white and one blue marble?
Hard (Difficulty Level-4):
26. Five A-list actresses are vying for the three leading roles in the new film, “Catfight in Denmark.” The actresses are Julia Roberds, Meryl Strep, Sally Fieldstone, Lauren Bake-all, and Hallie Strabwerry. Assuming that no actress has any special advantage over other in getting any role, what is the probability that Julia and Hallie will star in the film together?
27. A magician has five animals in his magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?
Very Hard & Tricky (Difficulty Level-5):
28. A florist has 2 azaleas, 3 buttercups, and 4 petunias. She puts two flowers together at random in a bouquet. However, the customer calls and says that she doesn’t want two of the same flower. What is the probability that the florist doesn’t have to change the bouquet?
29. If Lauren, Mary, Nancy, Oprah, and Penny sit randomly in a row, what is the probability that Opran and Penny do not sit next to each other?
30.At a certain pizza, 1⁄6 of the pizzas sold in a week were cheese, and 1⁄5 of the other pizzas sold were pepperoni. If Zach bought a pizza from the pizzeria that week, what is the probability that he ordered a pepperoni?
Answer Sheet:
1. 5C2 = 10
2. 1260
3. 3! × 3! = 36
4. 7C3 = 35
5. 10P5 or 10 × 9 × 8 × 7 × 6 = 30240
6. 5 × 4 = 20 = B
7. 4C2 + 4C3 + 4C4 = 11 = D
8. 9C1 × 6C2 × 3 = 405
9. 2C1 × 5C4 × 2C1 = 20
10. 9!⁄(5! × 4!) = 126
11. 3! × 3! + 3! × 3! = 72
12. 6P5 = 720 (As order matter)
13. 6C2 + 6C3 = 35
14. 6C3 × 15 = 300 minutes = 5 hours
15. (5!⁄2!) – (4!⁄2!) = 48
16. 2⁄9
17. D
18. A
19. 1 – (3⁄36) = 1 – (1⁄12) = 11⁄12
20. 1⁄36
21. 1⁄6
22. 1⁄3
23. 3⁄28
24. 0.8 × 0.8 × 0.2 = 0.128 = 12.8%
25. (2⁄10) × (5⁄9) × 2 = 2⁄9
26. 3⁄10
27. 2⁄5 = 40%
28. 13⁄18
29. 3⁄5 {As 1 – (4!⁄5! × 2)}
30. Simple 1⁄6 🙂 (It was simple!)