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05 May 2026, 00:24
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Difficulty:
75%
(hard)
Question Stats:
11% (01:50) correct
89%
(01:53)
wrong
based on 18
sessions
History
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Giuseppe’s offers a certain number of distinct toppings. Paco and Lucia each independently order a pizza with exactly 3 different toppings, chosen at random. The probability that their pizzas have exactly one topping in common is greater than 1/2.
Which of the following could be the number of toppings Giuseppe’s offers?
I. 9
II. 10
III. 11
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Which of the following could be the number of toppings Giuseppe’s offers?
I. 9
II. 10
III. 11
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
05 May 2026, 23:38
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Giuseppe’s offers a certain number of distinct toppings. Paco and Lucia each independently order a pizza with exactly 3 different toppings, chosen at random. The probability that their pizzas have exactly one topping in common is greater than 1/2.
Which of the following could be the number of toppings Giuseppe’s offers?
I. 9
We can start by saying that Paco chooses 3 toppings.
Now, we can determine the probability that Lucia chooses one of the same toppings. We can work it this way because all that matters is whether Lucia chooses exactly one of the same toppings. That's the same outcome as Lucia choosing and Paco choosing one of hers.
Total ways Lucia can choose = 9c3 = 84
Ways Lucia can choose one in common with Paco = 3c1 × 6c2 = 45
Probability that Lucia chooses one in common = 45/84 = 15/28
15/28 > 1/2
Keep.
II. 10
Total ways Lucia can choose = 10c3 = 120
Ways Lucia can choose one in common with Paco = 3c1 × 7c2 = 63
Probability that Lucia chooses one in common = 63/120 = 21/40
21/40 > 1/2
Keep.
III. 11
Total ways Lucia can choose = 11c3 = 165
Ways Lucia can choose one in common with Paco = 3c1 × 8c2 = 84
Probability that Lucia chooses one in common = 84/165 = 28/55
28/55 > 1/2
Keep.
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Correct answer: E
Which of the following could be the number of toppings Giuseppe’s offers?
I. 9
We can start by saying that Paco chooses 3 toppings.
Now, we can determine the probability that Lucia chooses one of the same toppings. We can work it this way because all that matters is whether Lucia chooses exactly one of the same toppings. That's the same outcome as Lucia choosing and Paco choosing one of hers.
Total ways Lucia can choose = 9c3 = 84
Ways Lucia can choose one in common with Paco = 3c1 × 6c2 = 45
Probability that Lucia chooses one in common = 45/84 = 15/28
15/28 > 1/2
Keep.
II. 10
Total ways Lucia can choose = 10c3 = 120
Ways Lucia can choose one in common with Paco = 3c1 × 7c2 = 63
Probability that Lucia chooses one in common = 63/120 = 21/40
21/40 > 1/2
Keep.
III. 11
Total ways Lucia can choose = 11c3 = 165
Ways Lucia can choose one in common with Paco = 3c1 × 8c2 = 84
Probability that Lucia chooses one in common = 84/165 = 28/55
28/55 > 1/2
Keep.
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Correct answer: E
