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
done, look back
To find the number of different types of tacos costing $2.25 that can be made at the stand, we need to determine the number of possible combinations of ingredients.
Since a taco can have any three ingredients for a flat price of $1.25, we can partition the $2.25 into two parts: $1.25 for the initial three ingredients and $1.00 for any additional ingredients.
Let's calculate the number of possible combinations for the initial three ingredients:
- We have ten different ingredients to choose from (since there are ten possible ingredients).
- We need to choose three ingredients to make the taco.
- The order of the ingredients does not matter.
In this case, we can use the combination formula, denoted as nCr, to calculate the number of different combinations. The formula to calculate nCr is:
nCr = n! / (r! * (n-r)!)
where n is the total number of items available and r is the number of items to be selected.
The combination formula can be simplified to:
nCr = n * (n-1) * (n-2) * ... * (n-r+1) / r!
For the initial three ingredients, we have:
n = 10 (number of ingredients)
r = 3 (number of ingredients to choose)
Using the combination formula, we can calculate:
10C3 = 10 * 9 * 8 / (3 * 2 * 1) = 120 / 6 = 20
So, there are 20 different combinations for the initial three ingredients.
Now, let's calculate the number of possible combinations for additional ingredients:
- We have seven remaining ingredients to choose from (since we used three ingredients for the initial selection).
- We can add any number of additional ingredients, including zero.
For each additional ingredient, we have two options: either include it or exclude it.
Using the concept of combinations with repetition, the number of combinations when choosing from seven ingredients with repetition allowed can be calculated using the formula:
(n + r - 1)Cr = (n + r - 1)! / (r! * (n-1)!)
where n is the total number of items available (in this case, seven ingredients), and r is the number of items to be selected (in this case, the number of additional ingredients).
For the additional ingredients, we have:
n = 7 (number of remaining ingredients)
r = 0, 1, 2, ..., 7 (number of additional ingredients)
Using the formula, we can calculate the number of combinations for each value of r:
7C0 = 1
7C1 = 7
7C2 = 28
7C3 = 84
7C4 = 210
7C5 = 462
7C6 = 924
7C7 = 1716
Now, let's multiply the number of combinations for the initial three ingredients (20) by the number of combinations for each value of r (1, 7, 28, 84, 210, 462, 924, 1716):
20 * (1 + 7 + 28 + 84 + 210 + 462 + 924 + 1716) = 20 * 4432 = 88,640
So, there are 88,640 different types of tacos costing $2.25 that can be made at the stand.
Since none of the answer choices match 88,640, it seems that there may be a mistake in the question, answer choices, or calculations.