How many different ways can you choose 4 toppings from a selection of 10?

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Question: How many different ways can you choose 4 toppings from a selection of 10?

Options:

Correct Answer: 210

Solution:

The number of combinations of 10 toppings taken 4 at a time is C(10, 4) = 210.

How many different ways can you choose 4 toppings from a selection of 10?

How many different ways can you choose 4 toppings from a selection of 10?

Step 1: Understand that we want to choose 4 toppings from a total of 10 toppings.
Step 2: Recognize that the order in which we choose the toppings does not matter. This means we will use combinations, not permutations.
Step 3: The formula for combinations is C(n, r) = n! / (r! * (n - r)!), where n is the total number of items to choose from, r is the number of items to choose, and '!' denotes factorial.
Step 4: In our case, n = 10 (the total toppings) and r = 4 (the toppings we want to choose).
Step 5: Plug the values into the formula: C(10, 4) = 10! / (4! * (10 - 4)!) = 10! / (4! * 6!).
Step 6: Calculate the factorials: 10! = 10 × 9 × 8 × 7 × 6!, 4! = 4 × 3 × 2 × 1 = 24, and 6! cancels out in the equation.
Step 7: Simplify the equation: C(10, 4) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
Step 8: Calculate the numerator: 10 × 9 × 8 × 7 = 5040.
Step 9: Calculate the denominator: 4 × 3 × 2 × 1 = 24.
Step 10: Divide the numerator by the denominator: 5040 / 24 = 210.
Step 11: Conclude that there are 210 different ways to choose 4 toppings from a selection of 10.

Combinations – This problem tests the understanding of combinations, specifically how to calculate the number of ways to choose a subset of items from a larger set without regard to the order of selection.