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mamma's pizza advertises a special where you can choose a thin crust, thick crust, or cheese crust pizza with any combination of toppings. the ad says that there are almost 200 ways that you can order your pizza. what is the smallest amount of toppings available?
mamma's pizza advertises a special where you can choose a thin crust, thick crust, or cheese crust pizza with any combination of toppings. the ad says that there are almost 200 ways that you can order your pizza. what is the smallest amount of toppings available?
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1) There are 3 ways to choose the type of crust (assuming that we choose exactly one of the three types of crust for the pizza). Also, for each available topping, we have 2 choices: either include or exclude that topping.
So if n is the number of toppings, then by the Fundamental Counting Principle,
number of ways to order the pizza = 3(2^n).
(No, the number of ways is not just 3n, since one can choose no toppings and can also choose more than one topping.)
Since there are "almost" 200 ways to order the pizza, 3(2^n) is "almost" 200.
Thus, 2^n is "almost" 66 2/3. Note that 2^6 = 64 and 2^7 = 128, so 6 toppings is the most reasonable answer. (For 6 toppings, there are 192 possible ways to order the pizza, which is "almost" 200.)
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2) We have three tasks: choose which order to place the group of men and the group of women (2 ways), choose how to arrange the 3 men within the group of men (3*2*1=6 ways), and choose how to arrange the 3 women within the group of women (3*2*1=6 ways).
By the Fundamental Counting Principle, there are 2*6*6 = 72 possible seating arrangements.
Lord bless you today!
So if n is the number of toppings, then by the Fundamental Counting Principle,
number of ways to order the pizza = 3(2^n).
(No, the number of ways is not just 3n, since one can choose no toppings and can also choose more than one topping.)
Since there are "almost" 200 ways to order the pizza, 3(2^n) is "almost" 200.
Thus, 2^n is "almost" 66 2/3. Note that 2^6 = 64 and 2^7 = 128, so 6 toppings is the most reasonable answer. (For 6 toppings, there are 192 possible ways to order the pizza, which is "almost" 200.)
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2) We have three tasks: choose which order to place the group of men and the group of women (2 ways), choose how to arrange the 3 men within the group of men (3*2*1=6 ways), and choose how to arrange the 3 women within the group of women (3*2*1=6 ways).
By the Fundamental Counting Principle, there are 2*6*6 = 72 possible seating arrangements.
Lord bless you today!
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Qn1
3 types of crust
n toppings
So you have 3n ways, and ad claimed atleast 200.
3n >= 200
n >= 66
Atleast 66 toppings
Qn2
3 men can be sat in 3! ways = 6
3 women can be sat in 3! ways = 6
So the total number of ways = 6 X 6 = 36.
_If_ it matters whether the men or women take the first seats then the total will be 2 * 36 = 72.
3 types of crust
n toppings
So you have 3n ways, and ad claimed atleast 200.
3n >= 200
n >= 66
Atleast 66 toppings
Qn2
3 men can be sat in 3! ways = 6
3 women can be sat in 3! ways = 6
So the total number of ways = 6 X 6 = 36.
_If_ it matters whether the men or women take the first seats then the total will be 2 * 36 = 72.