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01 Nov 2013, 18:34
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
15%
(low)
Question Stats:
81% (01:19) correct
19%
(01:36)
wrong
based on 593
sessions
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Mario's Pizza has two choices of crust: deep dish and thin-and-crispy. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. Finally, Mario's offers very pizza in extra-cheese as well as regular. If Linda's volleyball team decides to order a pizza with four toppings, how many different choices do the teammates have at Mario's Pizza?
A. 4
B. 5
C. 10
D. 15
E. 20
A. 4
B. 5
C. 10
D. 15
E. 20
Show SpoilerMy approach
Total number of choices = 2 crust x 5 toppings x 2 cheese = 20
Vball team: 2 crust x 4 toppings x 2 cheese = 16 choices
I understand this approach is incorrect because instead of first saying choose 4 out of 5 toppings, it essentially eliminates a topping.
However, why would the correct approach, 2 crust x 5c4 toppings x 2 cheese, be the same as the "total" possibilities, 20? Is it just a coincidence that the numbers worked out this way in this problem?
Vball team: 2 crust x 4 toppings x 2 cheese = 16 choices
I understand this approach is incorrect because instead of first saying choose 4 out of 5 toppings, it essentially eliminates a topping.
However, why would the correct approach, 2 crust x 5c4 toppings x 2 cheese, be the same as the "total" possibilities, 20? Is it just a coincidence that the numbers worked out this way in this problem?
15 Jun 2015, 04:51
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apple08
There is one more piece of information: Mario's offers very pizza in extra-cheese as well as regular.
Mario's Pizza has two choices of crust: deep dish and thin-and-crispy. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. Finally, Mario's offers very pizza in extra-cheese as well as regular. If Linda's volleyball team decides to order a pizza with four toppings, how many different choices do the teammates have at Mario's Pizza?
Mario's Pizza:
2 choices of crust.
5 choices of toppings.
2 choices of cheese (extra-cheese or regular).
Linda decides to order a pizza with four toppings.
How many 4-topping pizzas are possible from 5? \(C^4_5=5\). But the pizza can have 2 different crusts and two different cheese, therefore the final answer is 5*2*2 = 20.
General Discussion
01 Nov 2013, 22:15
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Yes, it is a coincidence that you arrived at the correct answer on this problem. It's unfortunate when that happens, because it's often hard to figure out why the approach is wrong.
Basically, your middle term (5 toppings) is appropriate for selecting 1 topping at a time out of 5 options.
Coincidentally, this parallels the correct approach of selecting 4 toppings at a time out of 5 options. With the latter, you effectively leave out 1 topping for each option, which has the effect of also giving you 5 different combinations.
But notice, having 5 different groupings of 4 toppings each (5c4) is very different from choosing 1 of 5 toppings at a time (5c1). They just happen to provide the same number of possible configurations.
Basically, your middle term (5 toppings) is appropriate for selecting 1 topping at a time out of 5 options.
Coincidentally, this parallels the correct approach of selecting 4 toppings at a time out of 5 options. With the latter, you effectively leave out 1 topping for each option, which has the effect of also giving you 5 different combinations.
But notice, having 5 different groupings of 4 toppings each (5c4) is very different from choosing 1 of 5 toppings at a time (5c1). They just happen to provide the same number of possible configurations.
