What is a combination?

A combination is a selection of items from a larger set where the order of selection does not matter. For example, choosing 2 fruits from a set of 3 (apple, banana, cherry) can be done in 3 ways: (apple, banana), (apple, cherry), or (banana, cherry).

How do you calculate the number of combinations of n items taken r at a time?

The number of combinations of n items taken r at a time is calculated using the formula: C(n, r) = n! / (r! * (n - r)!), where '!' denotes factorial, meaning the product of all positive integers up to that number.

Calculate the number of combinations of 5 items taken 2 at a time.

Using the formula C(5, 2) = 5! / (2! * (5 - 2)!) = 5! / (2! * 3!) = (5 × 4) / (2 × 1) = 10. Thus, there are 10 combinations.

If a committee of 4 is to be formed from a group of 10 people, how many different committees can be formed?

Hint: Use the combination formula.

Using the combination formula: C(10, 4) = 10! / (4! * (10 - 4)!) = 10! / (4! * 6!) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 210. Therefore, 210 different committees can be formed.

What is the value of C(7, 0)?

C(7, 0) = 7! / (0! * (7 - 0)!) = 7! / (1 * 7!) = 1. This means there is exactly one way to choose 0 items from 7, which is to choose nothing.

If you have 8 books and want to choose 3 to take on vacation, how many different selections can you make?

Hint: Apply the combination formula C(n, r).

Using the formula: C(8, 3) = 8! / (3! * (8 - 3)!) = (8 × 7 × 6) / (3 × 2 × 1) = 56. Therefore, you can make 56 different selections.

In how many ways can you select a president and a vice president from a club of 5 members?

Hint: This involves permutations, not combinations.

Since the order matters (president vs vice president), use permutations. P(5, 2) = 5! / (5 - 2)! = 5 × 4 = 20. Thus, there are 20 ways.

How many ways can you choose 2 toppings from a list of 6 for your pizza?

Hint: Use the combination formula C(n, r).

Using the formula: C(6, 2) = 6! / (2! * (6 - 2)!) = (6 × 5) / (2 × 1) = 15. Therefore, you can choose 15 different combinations of toppings.