Combination without Repetition

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In the standard Combination case, no repeated elements are allowed, so you need to choose unique items from a group.

Combination Example
Due to budget cuts, there will only be 1 winner in this year's poetry contest. Luckily only 3 people (Anna, Bill and Charlie) entered the contest. How many ways can you choose 1 winner from 3 contestants?

There are 3 possible choices. (Duh.)

Real Example
The poetry club raised some more money and can now award 2 identical prizes (without differentiating between 1^st and 2^nd place). How many ways can you choose 2 winners from 3?

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If it was a permutation, you would just do:
3 * 2 = 6

You could list the 6 possible permutations:


{Anna, Bill}	{Bill, Anna}
{Anna, Charlie}	{Charlie, Anna}
{Bill, Charlie}	{Charlie, Bill}

However, selections with the same elements (in a different order) should count as 1 combination. Choosing {Anna, Bill} is the same as {Bill, Anna}. As you can see, there are 2 permutations for each combination. This is because there are 2 ways to order every 2-element combination.

Challenge
Great news! The poetry budget increased, and you now need to award 3 identical prizes (They still don't have enough money for different prizes for 1^st, 2^nd and 3^rd place) . When the news spread about the prices, more people signed up, and there are now 10 contestants. How many ways can you choose 3 winners?

Guideline
You already can calculate (10 Permute 3), the number of possible permutations. Try to figure out how many permutations p, there will be for each combination, and use that to get the final answer. The next Node will discuss the general formula.

Challenge

How many ways can you choose 3 winners from 10 contestants?

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