There are six cars in a motorcade. How many different arrangements of cars in the motorcade are possible?
-
Solution
The order of cars matters, so you simply need to multiply the number of possible cars for each spot. For the first car, there are six possible, for the second, five, and so on. So your scratch paper should look like this: 6 5 4 3 2 1, which is equivalent to 6!, or 720.
For her Halloween display, Margaret plans to arrange a row of alternating witch and ghost figurines. The row must begin with a witch figurine and end with a ghost figurines. Margaret plans to purchase either three of each type of figurine or four of each type of figurines, and each figurine will look unique. Depending on how many figurine she purchases, which of the following could be the number of ways that she could arrange her display?
Indicate all possible values.
-
Solution
C and E
This problem is about permutations, because the order of the figurines matter since they each look unique.You have two cases to consider here. Start with the option that Margaret buys three witches and three ghosts. In that case, she has 3 choices of witch for the first spot, 3 choices of ghost for the second spot, 2 choices of witch for the third spot, 2 choices of ghost for the fourth spot, 1 choice of witch for the fifth spot, and 1 choice of ghost for the sixth spot: Hence, her total umber of arrangements is 3 × 3 × 2 × 2 × 1 × 1 = 36, which is choice (C). If she buys four of each figurine, her number of arrangements is 4 × 4 × 3 × 3 × 2 × 2 × 1 × 1 = 576, choice (E).
Kate and Chad are planning their wedding dinner and must select 3 of 12 entrees and 2 of 3 desserts for their guests to be able to choose from. How many different combinations of offerings are possible?
-
Solution
660
The order of entrees the guests chose from does not matter.Therefore, the number of different possibilities of entrees is \(\frac{12\times 11\times 10}{3\times 2\times 1}\)=220.The order of desserts does not matter either, so there are \(\frac{12\times 11\times 10}{3\times 2\times 1}\)=3 possible different combinations of desserts. Multiplying the two together gives a total of 660 possible different combinations of offerings.
Esteban’s restaurant offers a lunch special. A customer can order a platter consisting of four different small dishes from a selection of twelve choices. How many different platters can a customer create?
-
Solution
This problem is about combinations, because the order of the dishes does not matter. Since you’re choosing 4 dishes, start by drawing 4 blanks. On top, write the number of choices: 12 choices for the first dish, then 11, 10, and 9. On the bottom, start with the size of the smaller group and count down: 4, 3, 2, and 1.Cancel the numbers on the bottom, and the numbers on top will multiply to 495.
Of a group of 10 PTA members, a committee will be selected that has 1 president and 3 other members. How many different committees could be selected?
-
Solution
840
There are 10 possible presidents. After the president is selected, there are 9 members left to fill the remaining 3 spots. Order does not matter, so the number of possibilities for the other three spots is \(\frac{9\times 8\times 7}{3\times 2\times 1}\). Simplifying the fraction yields 3 × 4 × 7 = 84. So, there are 10 possible presidents and 84 possible committees for each president. Multiplying them yields the total number of possible committees, 840.
Jess has nine different statues and chooses three to arrange in a display. How many arrangements can she make?
-
Solution
504
This problem deals with permutations because the order of the statues matters. Draw three slots for the three positions.You can choose from nine statues for the first spot, eight for the second, and seven for the third. Multiplying these values together gives you 504.
If Jeff has four movies, and must choose to watch either 1, 2, or 3 different movies, which of the following represents a possible number of different arrangements of movies that Jeff could watch?
Indicate all possible values.
-
Solution
A,E, and F
If Jeff watches one movie, he has four different choices for that one movie, so choice (A) is a correct answer.To find the total number of arrangements of two movies, first write out two slots. For the first movie, he has 4 choices and a 4 goes in the first slot. For the second movie, he now has three choices and a 3 goes in the second slot. 4 × 3 = 12, so choice (E) is correct.There are 24 arrangements if he watches three movies: 4 × 3 × 2 = 24.Choice (E) is also correct.
Six state governors meet at an annual convention. They line up in random order to pose for a photograph. If the governors of Alaska and Hawaii are among the six governors, how many different ways can the governors line up for the picture so that these two governors are adjacent?
-
Solution
This is a permutation because order matters. First, think about the positions for the 2 governors from Alaska and Hawaii.There are 5 pairs of spots they can occupy: first and second, second and third, third and fourth, fourth and fifth, or fifth and sixth.That gives you 5 possibilities; since either governor could come first, you have a total of 5 × 2 = 10 possible ways to arrange those 2 governors.Meanwhile, for each of those options, the other governors can assume any of the remaining spots, which equals 4 × 3 × 2 × 1, or 24 possibilities.The answer is thus 10 × 24 = 240, choice (D).
Depending on the day, an ice cream shop offers seven to nine possible ice cream flavors and three to four possible sauces. A Deluxe Sundae consists of two different types of ice cream and two different sauces. Which of the following could be the number of different Deluxe Sundaes that Eva can order?
Indicate all possible values.
-
Solution
C and D
Simplify this problem by dealing with the two combinations separately.To select 2 out of 7, 8, or 9 ice creams, calculate \(\frac{7}{2}\times \frac{6}{1},\frac{8}{2}\times \frac{7}{1}\), and \(\frac{9}{2}\times \frac{8}{1}\) to yield 21, 28, or 36 possible combinations of ice creams, respectively. Now, so the same thing for sauces:\(\frac{9}{2}\times \frac{8}{1}\)= 3, and \(\frac{9}{2}\times \frac{8}{1}\)= 6, so you have 3 or 6 possible combinations of sauces.The possible numbers of different Deluxe Sundaes, then, are 21 × 3 = 63; 21 × 6 = 126; 28 × 3 = 84; 28 × 6 = 168; 36 × 3 = 108; and 36 × 6 = 216. Only choices (C) and (D) work.
Depending on the night, a pizza restaurant offers anywhere from seven to nine different choices of toppings. Sam wants a pizza with three toppings. Which of the following could be the number of different ways that Sam can order his pizza with three different toppings?
Indicate all possible values.
-
Solution
A,B, and D
This problem is about combinations, because order doesn’t matter. On a night when the pizza place offers only seven toppings, Sam has (7 × 6 × 5) ÷ (1 × 2 × 3) = 35 options, choice (A). When the pizza place has eight toppings, Sam has (8 × 7 × 6) ÷ (1 × 2 × 3) = 42 options, choice (B). And when the pizza place has nine toppings, Sam has (9 × 8 × 7) ÷ (1 × 2 × 3) = 84 options, choice (D).