A video store has 8 new releases this week. Each is available on videotape and on DVD. How many ways can a customer choose a new release and a format to rent?
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How many ways can six different books be arranged on a shelf if one of the books is a dictionary and it must be on an end?
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For a college application, Macawi must select one of five topics on which to write a short essay. She must also select a different topic from the list for a longer essay. How many ways can she choose the topics for the two essays?
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An ice cream shop offers a choice of two types of cones and 15 flavors of ice cream. How many different 1-scoop ice cream cones can a customer order?
To find the answer to each of these questions, we can use the concept of counting principles or combinations.
1. For the first question, we need to find out the number of ways a customer can choose a new release and a format to rent. Since there are 8 new releases and each release is available on videotape and DVD, we can use the multiplication principle.
Number of ways = Number of options for new release * Number of options for format
Number of ways = 8 new releases * 2 formats
Number of ways = 16 ways
Therefore, a customer can choose a new release and a format to rent in 16 different ways.
2. In the second question, we need to know the number of ways six different books can be arranged on a shelf if one book, the dictionary, must be on an end. We can use the concept of permutations.
Number of ways = (Number of ways to arrange remaining five books) * (Number of ways to place the dictionary on an end)
Number of ways = 5! * 2!
Number of ways = 5 * 4 * 3 * 2 * 1 * 2
Number of ways = 120
Therefore, six different books can be arranged on a shelf with the dictionary on an end in 120 different ways.
3. For the third question, we need to find out how many ways Macawi can choose the topics for her two essays. Since she must choose one topic from a list of five topics for her short essay and a different topic for her longer essay, we can use the concept of combinations.
Number of ways = Number of options for short essay * Number of options for longer essay
Number of ways = 5 topics * 4 topics
Number of ways = 20
Therefore, Macawi can choose the topics for her two essays in 20 different ways.
4. In the last question, we need to determine the number of different 1-scoop ice cream cones a customer can order given a choice of two types of cones and 15 flavors of ice cream. We can again use the multiplication principle.
Number of ways = Number of options for cone * Number of options for ice cream flavor
Number of ways = 2 types of cones * 15 flavors of ice cream
Number of ways = 30
Therefore, a customer can order a different 1-scoop ice cream cone in 30 different ways.