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12 Jun 2025, 11:48
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A pizza restaurant has 2 choices of crust: deep dish crust or thin crust. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. The order of the toppings doesn’t matter. Finally, the restaurant offers every pizza in extra cheese as well as regular. If someone orders a pizza with 4 unique toppings, how many different combinations are possible for the toppings?
19 Jun 2025, 12:24
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Set up 4 slots for the toppings and label all the slots the same, since order doesn’t matter. Fill in the numbers from 5 on down, because the person orders 4 unique toppings.
Now multiply the numbers and divide by 4! because the labels are all the same.
Alternatively, use the combinations formula to count the combinations of toppings:
Or use an intuitive approach: choosing 4 toppings out of 5 is equivalent to choosing the 1 topping that will not be on the pizza. There are only 5 ways to reject a single topping.
Next, since each of these pizzas can also be offered in 2 choices of crust, there are 5 × 2 = 10 possible pizzas with 4 unique toppings and one of the two crusts.
The same logic applies for extra cheese and regular. The number of pizzas with 4 toppings, one crust, and one version of cheese is 5 × 2 × 2 = 20.
GMAT-Club-Forum-t8gojh43.png [ 7.46 KiB | Viewed 86 times ]
GMAT-Club-Forum-eznom4zr.png [ 8.26 KiB | Viewed 87 times ]
GMAT-Club-Forum-4633l2tk.png [ 5.56 KiB | Viewed 85 times ]
Set up 4 slots for the toppings and label all the slots the same, since order doesn’t matter. Fill in the numbers from 5 on down, because the person orders 4 unique toppings.
Now multiply the numbers and divide by 4! because the labels are all the same.
Alternatively, use the combinations formula to count the combinations of toppings:
Or use an intuitive approach: choosing 4 toppings out of 5 is equivalent to choosing the 1 topping that will not be on the pizza. There are only 5 ways to reject a single topping.
Next, since each of these pizzas can also be offered in 2 choices of crust, there are 5 × 2 = 10 possible pizzas with 4 unique toppings and one of the two crusts.
The same logic applies for extra cheese and regular. The number of pizzas with 4 toppings, one crust, and one version of cheese is 5 × 2 × 2 = 20.
Attachment:
GMAT-Club-Forum-t8gojh43.png [ 7.46 KiB | Viewed 86 times ]
Attachment:
GMAT-Club-Forum-eznom4zr.png [ 8.26 KiB | Viewed 87 times ]
Attachment:
GMAT-Club-Forum-4633l2tk.png [ 5.56 KiB | Viewed 85 times ]
