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Check: Using the formula, \dfrac{11!}{(11-3)!} = \dfrac{11!}{8!} = 11 \times 10 \times 9 = 990 ✓
Tip: Different titles or positions (1st/2nd/3rd, president/secretary) mean order matters — use a permutation. Only use a combination when the chosen people are all equal.
Question 2 of 10 · Counting, combinations and permutations
A frozen-yogurt shop offers 9 different toppings. A "Pick 3" cup lets you choose any 3 different toppings.
How many different "Pick 3" cups are possible?
Hint
Is a strawberry-granola-almond cup different from an almond-granola-strawberry cup?
Step-by-step solution
The order of the toppings doesn't change the cup, so order does not matter → combination.
Use {}_nC_r = \dfrac{{}_nP_r}{r!} with n = 9, r = 3.
Permutations: 9 \times 8 \times 7 = 504.
Each group of 3 toppings can be ordered in 3! = 6 ways, so divide: {}_9C_3 = \frac{504}{6} = 84
There are 84 different cups.
Check: On the TI-30XS, 9 nCr 3 gives 84 ✓
Tip: When order doesn't matter, first count as if it did, then divide by r! to remove the duplicate orderings.
Question 3 of 10 · Counting, combinations and permutations
A café's breakfast combo includes one drink and one main dish. For the drink you may choose any of 5 hot drinks or 2 juices. There are 6 main dishes.
How many different breakfast combos are possible?
Hint
First count how many drinks there are in all. Is a hot drink chosen and a juice, or a hot drink or a juice?
Step-by-step solution
The drink is one item: a hot drink or a juice. "Or" inside one stage means add: 5 + 2 = 7 \text{ drinks}
Then choose a drink and a main dish. "And" between stages means multiply: 7 \times 6 = 42
Each set of 5 numbers can be ordered in 5! = 120 ways, so divide:
\frac{1{,}860{,}480}{120} = 15{,}504
There are 15,504 possible sets.
Check: On the TI-30XS, 20 nCr 5 = 15,504 ✓. Also 15{,}504 \times 120 = 1{,}860{,}480 ✓
Tip: Lottery picks, committees, and hands of cards are combinations. Only if the numbers had to match in the exact order would it be a permutation.
Question 6 of 10 · Probability of simple and compound events
Fill in
A café surveyed 160 customers about their favorite drink.
Coffee
Tea
Total
Under 30
33
27
60
30 and over
67
33
100
Total
100
60
160
One of the customers who prefer tea is chosen at random. What is the probability that this customer is under 30? Write your answer as a decimal.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The person is chosen only from the tea drinkers. What is the total of that group?
Step-by-step solution
The person is chosen from the tea drinkers only, so the total is the Tea column total: 60.
Favorable: tea drinkers who are under 30 = 27.
P(\text{under 30, given tea}) = \frac{27}{60}
Simplify: \dfrac{27}{60} = \dfrac{9}{20} = 0.45.
The probability is 0.45 (45%).
Check: The tea drinkers 30 and over give \tfrac{33}{60} = 0.55, and 0.45 + 0.55 = 1 ✓
Tip: Watch the phrase "of the ___" or "who ___": it tells you which row or column total to divide by. Dividing by the grand total (\tfrac{27}{160} \approx 0.17) answers a different question — "under 30 and prefers tea."
Question 7 of 10 · Probability of simple and compound events
Fill in
A standard deck has 52 cards in 4 suits (hearts, diamonds, clubs, spades) of 13 cards each. Each suit has 3 face cards (jack, queen, king).
One card is drawn at random. What is the probability that it is a heart or a face card? Round to the nearest hundredth.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Some cards are both a heart and a face card. How many? Make sure you count them only once.
Check: The flu column splits into 12 + 30 = 42 ✓, and \tfrac{12}{42} + \tfrac{30}{42} = 1 ✓
Tip: In a two-way table, the "given" group decides the denominator. Circle that row or column and ignore the rest of the table.
Question 10 of 10 · Counting, combinations and permutations
A burrito stand offers 4 tortillas, 6 fillings, and 3 salsas. A burrito has one of each. The owner will add one new item to the menu: a new tortilla, a new filling, or a new salsa.
Which addition would increase the number of possible burritos the most?
Hint
Find the number of burritos now. Then recompute it three times, adding 1 to a different list each time.
Step-by-step solution
Now: 4 \times 6 \times 3 = 72 burritos.
New tortilla: 5 \times 6 \times 3 = 90 → an increase of 90 - 72 = 18.
New filling: 4 \times 7 \times 3 = 84 → an increase of 84 - 72 = 12.
New salsa: 4 \times 6 \times 4 = 96 → an increase of 96 - 72 = 24.
The largest increase comes from a new salsa (+24).
Check: A new item in one list pairs with every combination from the other two lists. New salsa: 4 \times 6 = 24 new burritos ✓; new tortilla: 6 \times 3 = 18 ✓; new filling: 4 \times 3 = 12 ✓
Tip: Adding one item to the smallest list helps most, because that item pairs with the product of the larger lists.