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8-6 Combinations8-6 Combinations

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CombinationsCombinationsRemember that Permutations told us how Remember that Permutations told us how

many different ways we could choose r many different ways we could choose r items from a group of n items if the order items from a group of n items if the order mattered.mattered.

The formula we used was:The formula we used was:

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CombinationsCombinationsCombinations tell us how many ways we can pick r Combinations tell us how many ways we can pick r

items from a group of n items if the order doesn’t items from a group of n items if the order doesn’t matter. matter.

For example: If 10 people enter a contest where the For example: If 10 people enter a contest where the first prize is $100, 2first prize is $100, 2ndnd is $75, and 3 is $75, and 3rdrd is $50, then is $50, then order matters – this is Permutations:order matters – this is Permutations:

If however, the three top prizes are $50 each, then If however, the three top prizes are $50 each, then the order doesn’t matter – this is now referred to the order doesn’t matter – this is now referred to as Combinations. as Combinations.

Of the Three who finish, there are 6 different ways Of the Three who finish, there are 6 different ways they can be arranged, but they are all the same they can be arranged, but they are all the same combination of a winning set of people.combination of a winning set of people.

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Combinations Formula:Combinations Formula:

Are there more Combinations or Permutations?Are there more Combinations or Permutations?

So, each combination of r items can be arranged in So, each combination of r items can be arranged in r! ways:r! ways:

Ex: 20 distinct points areEx: 20 distinct points are

chosen on a circle.chosen on a circle.

a) How many segments cana) How many segments can

be formed?be formed?

b) How many Triangles canb) How many Triangles can

be formed?be formed?

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Examples:Examples:At a restaurant, you can order pizza with any of 9 At a restaurant, you can order pizza with any of 9

different toppings. How many different pizzas different toppings. How many different pizzas are possible with exactly 3 of these toppings?are possible with exactly 3 of these toppings?

Five students from your class are randomly picked Five students from your class are randomly picked to be interviewed by the local newspaper. If to be interviewed by the local newspaper. If your class contains 20 students, what is the your class contains 20 students, what is the probability that neither you nor your best friend probability that neither you nor your best friend will be picked?will be picked?

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