A taco stand sells tacos with any three ingredients for a flat price of $1.25. Additional ingredients cost $0.50 each. There are ten different possible ingredients. How many different types of tacos costing $2.25 can be made at the stand?

A. 160
B. 252
C. 345
D. 452
E. 610

to cost $2.25 there must have been 5 ingredients on each of the tacos.

number of ways to choose 5 from 10
= C(10,5) = 252

To find the number of different types of tacos costing $2.25, we need to consider the number of possible combinations of ingredients.

Since there are ten different possible ingredients, we can choose any three of them for the initial flat price of $1.25. For this selection of three ingredients, there are 10 choose 3 (written as C(10,3)) possible combinations.

After choosing the initial three ingredients, we can add additional ingredients to the taco. Since each additional ingredient costs $0.50, the number of additional ingredients we can choose depends on the remaining budget.

To find the number of additional ingredients, we need to subtract the initial flat price from the total budget:
Remaining Budget = $2.25 - $1.25 = $1.00

Since each additional ingredient costs $0.50, we need to divide the remaining budget by $0.50 to find the number of additional ingredients:
Number of Additional Ingredients = Remaining Budget / $0.50

Now, we can calculate the total number of different types of tacos by multiplying the number of possible combinations of initial ingredients by the number of possible combinations of additional ingredients:
Total Number of Different Tacos = C(10,3) * (Number of Additional Ingredients + 1)

Calculating the above expression will give us the answer. Let's calculate it:

Number of Additional Ingredients = $1.00 / $0.50 = 2

Total Number of Different Tacos = C(10,3) * (2 + 1)
Total Number of Different Tacos = 120 * 3
Total Number of Different Tacos = 360

Therefore, the answer is not among the given options.