Question

Consider the following scenario, and find the resulting combinations

a) You are eating at a restaurant that has 10 dessert options - how many different sets of three desserts could you order?
b) You are eating at a restaurant that has 10 dessert options and your friend asks you to rank your top three desserts - how many different ways could you rank these desserts?

Answer 1

There are 10 different sets of three desserts that could be ordered and 720 different ways to rank the top three desserts.

Step-by-step explanation:

In order to find the number of different sets of three desserts that could be ordered from a restaurant with 10 options, we can use the concept of combinations. The formula for combinations is given by nCr = n! / (r! * (n-r)!), where n is the total number of options and r is the number of options to choose. In this case, n = 10 and r = 3.

Substituting the values into the formula, we have 10C3 = 10! / (3! * (10-3)!) = 120 / (6 * 7) = 120 / 42 = 10.

Therefore, there are 10 different sets of three desserts that could be ordered.

To find the number of different ways to rank the top three desserts, we can use the concept of permutations. The formula for permutations is given by nPr = n! / (n-r)!, where n is the total number of options and r is the number of options to choose. In this case, n = 10 and r = 3.

Substituting the values into the formula, we have 10P3 = 10! / (10-3)! = 10! / 7! = 10 * 9 * 8 = 720.

Therefore, there are 720 different ways to rank the top three desserts.

Answer 2

To find the number of sets of three desserts from 10 options, one would use combinations with a result of 120 different sets. To rank the top three desserts, permutations are used, leading to 720 different rankings.

Step-by-step explanation:

The question involves finding the number of combinations and permutations of dessert options at a restaurant, which is a topic in mathematics, specifically combinatorics.

1. To find the number of different sets of three desserts that could be ordered from 10 dessert options without regard to the order, we use combinations. The formula for combinations is given by C(n, k) = n! / (k! * (n-k)!), where n is the total number of items to choose from, and k is the number of items to pick. Therefore, the number of combinations of 10 desserts taken 3 at a time is C(10, 3) = 10! / (3! * (10-3)!) = 120 possible sets.
2. To rank your top three desserts out of 10, where the order matters, we use permutations. The formula for permutations is P(n, k) = n! / (n-k)!. Thus, the number of ways to rank three desserts out of ten is P(10, 3) = 10! / (10-3)! = 720 different rankings.