Question

a group of five friends goes to a restaurant for dinner. the restaurant offers 20 different main dishes. (a) suppose that the group collectively selects five different dishes to share. the waiter just needs to place all five dishes in the center of the table. how many different possible meals are there for the group? (b) suppose that each individual selects a main course. the waiter must remember who selected which dish. it's possible for more than one person to select the same dish. how many different possible meals are there for the group? (c) suppose that each individual selects a main course. the waiter must remember who selected which dish. however, the friends agree that no two people will select the same dish. how many different possible meals are there for the group?

Answer 1

a) The number of different possible meals when the group collectively selects five different dishes is C(20, 5). b) The number of different possible meals when each individual selects a main course and it's possible for more than one person to select the same dish is 20^5. c) The number of different possible meals when each individual selects a main course and no two people will select the same dish is 20 * 19 * 18 * 17 * 16.

Step-by-step explanation:

(a) To find the number of different possible meals for the group when they collectively select five different dishes, we can use the concept of combinations. Since there are 20 main dishes and the group needs to select 5, we can use the formula for combinations: C(n, r) = n! / (r! * (n-r)!), where n is the total number of options and r is the number of choices. So, the number of different possible meals is C(20, 5) = 20! / (5! * (20-5)!).

(b) To find the number of different possible meals when each individual selects a main course and it's possible for more than one person to select the same dish, we can use the concept of permutations with repetition. Since there are 20 main dishes and each person can choose from the same set of dishes, the number of different possible meals is 20^5.

(c) To find the number of different possible meals when each individual selects a main course and no two people will select the same dish, we can use the concept of permutations. Since there are 20 main dishes available and the first person can choose from all 20, the second person can choose from 19 remaining options, the third person can choose from 18 remaining options, and so on. So, the number of different possible meals is 20 * 19 * 18 * 17 * 16.