Combinations, Permutations, and Counting Principles

Front

What is the difference between combinations and permutations?

Back

Combinations refer to the selection of items without regard to the order, while permutations refer to the arrangement of items where the order matters.

Front

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

Back

The formula for combinations is C(n, r) = n! / (r!(n - r)!), where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.

Front

What is the formula for permutations of n items taken r at a time?

Back

The formula for permutations is P(n, r) = n! / (n - r)!, where n is the total number of items and r is the number of items to arrange.

Front

What are counting principles in combinatorics?

Back

Counting principles are rules that help determine the number of ways to arrange or combine items, including the addition and multiplication principles.

Front

How many different ways can you arrange 3 books on a shelf?

Back

The number of arrangements is 3! = 6.

Front

If you have 5 shirts and 3 pairs of pants, how many different outfits can you create?

Back

Using the multiplication principle, you can create 5 * 3 = 15 different outfits.

Front

What is the significance of the factorial function in permutations and combinations?

Back

The factorial function (n!) is used to calculate the total arrangements of n items and is fundamental in both permutations and combinations.

Front

How do you calculate the number of unique passcodes if digits and letters cannot be repeated?

Back

For a passcode of 2 digits and 2 letters, calculate as follows: 10 options for the first digit, 9 for the second, 26 for the first letter, and 25 for the second letter. Total = 10 * 9 * 26 * 25 = 58,500.

Front

What is the total number of ways to choose 3 people from a group of 10?

Back

Using combinations, the total is C(10, 3) = 10! / (3!(10 - 3)!) = 120.

Front

What is the multiplication principle in counting?

Back

The multiplication principle states that if one event can occur in m ways and a second can occur independently in n ways, then the two events can occur in m * n ways.