# Permutations & Combinations Calculator

An element is what makes up a set. Given a set of colors, blue may be an element of the set, or given a set of numbers '2' may be an element of the set.

The universe of all possible elements is the set itself.

Given the numbers 1, 2, 3, 4, 5, 6 then the universe of all elements is 1 - 6, and 4 is a single element from the universe. (A set is normally written this way: {1, 2, 3, 4, 5, 6} or possibly this way {1, 2, 3 ... 100}.)

Or given the colors red, blue green and yellow, the universe of all elements is red, blue, green and yellow. Red and blue are both elements in the set.

The combination is the number of different arrangements of elements when ordering is not important. An arrangement of blue/red is the same as red/blue and would thus be counted as one combination. Additionally combinations do not permit the repeated use of an element within an arrangement.

If a set contains these elements {1,2,3} here are the three possible 2 element combinations:

• 1,2
• 1,3
• 2,3

In the above 1,1 is not a combination because repetition is not allowed and 1,2 and 2,1 contain the same elements and thus is counted as one combination.

Example: If you want to know the odds of picking the winning lottery ticket look at the number of combinations. If you have to pick seven numbers (the number of elements in the subset) out of thirty-six (number of elements in the universe), it does not matter if you pick 1,2,3,4,5,6,7 or 7,6,5,4,3,2,1. The order is not important. In this case, the number of combinations is 8,347,665 and thus the odds of picking the winning numbers (assuming you are buying one ticket) is 1 in 8,347,655.

A permutation is the number of arrangements and the order of the elements within an arrangement is important. An arrangement of blue/red is not the same as red/blue. These would be counted as two permutations. Like combinations, permutations do not permit the repeated use of an element within an arrangement.

If a set contains these elements {1,2,3} here are the six possible 2 element permutations:

• 1,2
• 1,3
• 2,1
• 2,3
• 3,1
• 3,2

Example: If you want to know how many bets you have to place to cover the possibilities of three horses placing first, second or third in a race, you need to look at the number of permutations. Here, ordering is important. It matters if you pick horse A as first and horse B as second or horse B as first and horse A as second.

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