In how many ways can 5 ice cream toppings be chosen from 12 available toppings?
60
To determine the number of ways to choose 5 ice cream toppings from the available 12 toppings, we can use the concept of combinations.
The formula to find the number of combinations is given by:
nCr = n! / (r!(n-r)!)
Where n represents the total number of items to choose from, r represents the number of items to be chosen, and the exclamation mark (!) denotes the factorial of a number (i.e., the product of all positive integers up to that number).
In this case, we have n = 12 (the total number of available toppings) and r = 5 (the number of toppings to be chosen). Plugging these values into the formula, we get:
12C5 = 12! / (5!(12-5)!)
Now, let's calculate the values involved:
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
5! = 5 x 4 x 3 x 2 x 1
(12-5)! = 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1
Now, substitute these values back into the formula:
12C5 = 12! / (5!(12-5)!)
= (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ((5 x 4 x 3 x 2 x 1)((7 x 6 x 5 x 4 x 3 x 2 x 1))
= (12 x 11 x 10 x 9 x 8) / (5 x 4 x 3 x 2 x 1)
= 792
Therefore, there are 792 ways to choose 5 ice cream toppings from 12 available toppings.