Question

a.) If there are 7 different movies playing in a theater, but you can only watch 4 of them. How many different groupings of movies can you watch? b.) How many different ways can 10 people be arranged in order into 5 spots?

Answer 1

To determine the number of different groupings of movies, combinations are used, resulting in 35 groupings. For arranging people into spots, permutations are needed, leading to 30,240 different arrangements.

Step-by-step explanation:

Combination and Permutation Problems

To solve these types of problems, we use the concept of combinations and permutations from the field of combinatorics.

Part A: Combinations

For the first part of the question, which involves choosing 4 movies out of 7, we use combinations since the order of watching the movies does not matter. The formula for combinations is C(n, k) = n! / (k! * (n-k)!), where n is the total number of items, k is the number we want to choose, and ! denotes factorial.

In this case, n = 7 and k = 4. So the number of combinations is C(7, 4) = 7! / (4! * (7-4)!) = 7! / (4! * 3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35 different groupings of movies.

Part B: Permutations

For the second part, involving the arrangement of 10 people into 5 spots, the order matters, so we use permutations. The formula for permutations is P(n, k) = n! / (n-k)!, where n is the total number of items and k is the number we want to arrange.

Here, n = 10 and k = 5. Thus, the number of different ways is P(10, 5) = 10! / (10-5)! = 10! / 5! = (10 * 9 * 8 * 7 * 6) = 30240 different arrangements.