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Worksheet: Using Combinations to Count Outcomes

Q1:

Determine the number of ways a team of 119 people can be chosen out of a group of 120.

Q2:

Determine the number of ways a team of 2 people can be chosen out of a group of 17.

Q3:

David has enough money to purchase only two books from the six he wants to read. Determine the number of ways he can choose them.

Q4:

How many ways can a committee of 3 freshmen and 4 juniors be formed from a group of 8 freshmen and 11 juniors?

Q5:

How many lines are determined by 112 randomly drawn points, where no three of which are collinear?

Q6:

How many lines are determined by 44 randomly drawn points, where no three of which are collinear?

Q7:

How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?

Q8:

How many unique ways can a string of Christmas lights be arranged from 9 red, 10 green, 6 white, and 12 gold color bulbs?

  • A 1 5 5 5 2 0
  • B 6 4 8 0
  • C 4 . 8 3 × 1 0 8
  • D 3 . 0 3 × 1 0 1 9
  • E 9 7 2 0

Q9:

A motorcycle shop has the following vintage motorcycles: 10 choppers, 6 bobbers, and 5 cafe racers. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 cafe racers for a weekend showcase given that all the motorcycles of each type are unique?

Q10:

Just-For-Kicks sneaker company offers an online customizing service. For each basic sneaker, a customer can choose whether or not to include each of 11 different customization options. How many ways are there to design a custom pair of sneakers?

Q11:

A wholesale T-shirt company offers the sizes small, medium, large, and extra large in organic or nonorganic cotton and in the following colors: white, black, gray, blue, or red. How many different T-shirts are there to choose from?

Q12:

Mason wants to place billboard advertisements in 15 of the 30 neighborhoods in the county. How many ways can he choose which 15 neighborhoods to advertise in?

Q13:

A car wash offers the following optional services: clear coat wax, triple foam polish, undercarriage wash, rust inhibitor, wheel brightener, air freshener, and interior shampoo. Given that any number of these can be added to the basic wash, how many wash options are possible?

Q14:

Natalie bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?

Q15:

A restaurant offers a set menu with four choices of appetizers, five choices for the main course, and three choices for dessert. Assuming somebody orders all three courses, how many different possible meals can they choose?

Q16:

Jacqueline goes away for the weekend. She takes with her five jumpers, three skirts, and two pairs of shoes. How many different outfits can she wear?

  • A10
  • B15
  • C8
  • D30

Q17:

An ice cream stall offers eight different flavors of ice cream, and their ice cream is available in three different sizes: small, medium, and large.

How many possible combinations of flavors and sizes are there?

Assuming John chooses his ice cream completely at random, what is the probability that he chooses a small strawberry ice cream?

  • A 1 2 4
  • B 3 2 4
  • C 1 8
  • D 1 1 2
  • E 1 1 1

Assuming John chooses his ice cream completely at random, what is the probability that he chooses neither chocolate nor a large ice cream?

  • A 1 3 2 4
  • B 5 1 2
  • C 7 1 2
  • D 2 3
  • E 1 1 2 4

Q18:

Emma wants to purchase a new car. There are 6 different options for the color, 4 different options for the interior, 5 different options for the engine size, and 2 different options for the fuel type. How many different options does Emma have overall for her new car?

Q19:

How many possible outcomes are there when a fair coin is tossed at the same time as a number cube is rolled?

Q20:

Use the Fundamental Counting Principle to determine the total number of possible outcomes in choosing a card from a deck of cards numbered 1 5 , 1 6 , 1 7 , , 3 6 and picking a day of the week.

Q21:

Use the Fundamental Counting Principle to find the total number of possible outcomes of tossing 7 coins and selecting one letter from the word PERMUTATION.

Q22:

Two distinct digits are chosen at random from the set { 3 , 5 , 7 , 8 , 9 } . What is the probability that both digits are odd?

  • A 1 1 2 0
  • B 1 3 2 0
  • C 1 2 2 5
  • D 3 5
  • E 4 5

Q23:

In a class of 67 students, 54 passed a quiz. If two students are chosen at random from the entire class, what is the probability that at least one of them did NOT pass the quiz?

  • A 4 7 7 7 3 7
  • B 1 3 6 7
  • C1
  • D 2 6 0 7 3 7
  • E 2 1 3