Combinations Formula: Counting the number of ways to choose r items from n items.

Summary

This video explains the concept of counting and probability when order doesn't matter, contrasting it with cases where order is important. The example of forming a team from a group of five people is used to illustrate 'choosing' versus 'picking.' When choosing, the order doesn't matter, so the total number of arrangements is divided by the number of ways the items could be ordered. The formula for combinations is introduced, which calculates the number of ways to choose 'R' items from 'N' without caring about the order, using the formula: N! / (R!(N-R)!).

Takeaways

• 😀 In counting and probability, the order of selection may or may not matter, depending on the context.
• 😀 In the context of team selection, when the order of selection doesn't matter, the process is referred to as 'choosing' rather than 'picking'.
• 😀 When creating a team of 3 people from 5 options, the order of selection isn't important, so we focus on combinations rather than permutations.
• 😀 The formula for calculating combinations when order doesn't matter is: n! / (r! * (n-r)!)
• 😀 The difference between 'pick' and 'choose': 'Pick' considers order, while 'choose' doesn't.
• 😀 If order mattered in the team selection, we'd use permutations. For example, picking 3 people out of 5 would be calculated as 5 * 4 * 3.
• 😀 To convert a permutation problem (where order matters) into a combination problem (where order doesn't matter), we divide by the number of possible reorderings of the selected items.
• 😀 The number of ways to reorder a set of 3 people is given by 3! = 3 * 2 * 1 = 6.
• 😀 The mathematical notation for combinations is usually expressed as 'n choose r' or written as (n r), which represents the number of ways to choose r items from a set of n items without considering the order.
• 😀 The final formula for combinations is: n! / (r! * (n - r)!), which accounts for both the selection of items and the elimination of reordering.

Q & A

What does the term 'choose' refer to in counting and probability problems?

The term 'choose' refers to situations where the order in which selections are made does not matter. For example, when assembling a team, the specific order of selection doesn't affect the outcome.

How does the concept of 'choose' differ from 'pick' in probability?

'Pick' is used when the order in which items are selected matters, such as in a password combination where the first, second, and third digits are important. In contrast, 'choose' is used when the order of selection doesn't matter, as in forming a team from a set of people.

In the example of forming a team from five people, why is the order of selection not important?

In the team selection example, the focus is on choosing a group of people, not on the order in which they are selected. This means that different orderings of the same team are considered the same, so the order of selection is irrelevant.

How is the number of ways to choose a team calculated when order does not matter?

The number of ways to choose a team is calculated by using the combination formula, which is the total number of ways to pick a set of items divided by the number of ways to reorder them. This formula is typically written as 'n choose r', or 'C(n, r)'.

What is the formula for calculating 'n choose r'?

The formula for 'n choose r' is given by: C(n, r) = n! / (r!(n - r)!), where n is the total number of items to choose from, r is the number of items to be chosen, and '!' denotes factorial.

What does the factorial (n!) represent in the 'n choose r' formula?

The factorial (n!) represents the product of all integers from 1 to n. It calculates the total number of ways to arrange all n items in a sequence.

Why do we divide by r! when calculating 'n choose r'?

We divide by r! to account for the different ways the r selected items can be reordered among themselves. Since order doesn't matter in 'choose', we remove the redundancy caused by different orderings of the same set of items.

What is the role of (n - r)! in the 'n choose r' formula?

The term (n - r)! in the denominator accounts for the fact that, after selecting r items, there are (n - r) items left that are not chosen, and the number of ways to arrange those remaining items is irrelevant in this context.

How does dividing by the number of possible reorderings (r!) affect the result in a 'choose' problem?

Dividing by r! ensures that multiple arrangements of the same group of items (in different orders) are counted as a single combination. This eliminates the overcounting of different permutations of the same selection.

Can you explain the 'pick' versus 'choose' terminology with an example?

Sure! For instance, if you're selecting three people from a group of five for a team and care about the order (e.g., who is selected first, second, or third), you would 'pick'. If the order doesn't matter, you'd 'choose'. So, 'pick' involves permutations, while 'choose' involves combinations.