Permutations and Combinations
See also: probability
Permutations and Combinations Calculator
The calculator below will calculate number of permutations or combinations.
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Permutation is an ordered arrangement of a number of elements of a set.
Mathematically, given a set with n numbers of elements, the number of permutations of size r is denoted by P(n,r) or nPr or nPr.
The formula is given by
| P(n,r) = nPr = nPr = | n! |
| (n - r)! |
where n! (n factorial) = n × (n-1) × (n-2) × ... × 1 and 0! = 1.
For example, given the set of letters {a,b,c} the permutations of size 2 (take 2 elements of the set) are {a,b}, {b,a}, {a,c}, {c,a}, {b,c}, and {c,b}. Please note that the order is important (i.e. {a,b} is considered different from {b,a}).
The number of permutations is 6.
| P(3,2) = 3P2 = 3P2 = | 3! |
| (3 - 2)! | |
| = | 3 × 2 × 1 |
| 1! | |
| = | 6 |
| 1 | |
| = | 6 |
Another example: How many different ways are there can 5 different books be arranged on the self?
Answer: Here, n = 5 and r = 5.
So, 5P5 = 5!/(5-5)! = 5!/0! = (5 × 4 × 3 × 2 × 1)/1 = 120.
As can be seen from the above example, when n = r, the formula for nPr = n!.
Combination is an unordered arrangement of a number of elements of a set.
Given a set with n numbers of elements, the number of combinations of size r is denoted by C(n,r) or nCr or nCr.
The formula is given by
| C(n,r) = nCr = nCr = | n! |
| r! (n - r)! |
where n! (n factorial) = n × (n-1) × (n-2) × ... × 1 and 0! = 1.
For example, given the set of letters {a,b,c} the combinations of size 2 (take 2 elements of the set) are {a,b}, {a,c}, and {b,c}. Please note that the order is not important (i.e. {b,a} is considered the same as {a,b}).
The number of combinations is 3.
| C(3,2) = 3C2 = 3C2 = | 3! |
| 2! (3 - 2)! | |
| = | 3 × 2 × 1 |
| 2 × 1 × 1! | |
| = | 6 |
| 2 | |
| = | 3 |
Another example: A basket contains an apple, an orange, a pear, and a banana. How many combinations of three fruits are there?
Answer: Here, n = 4 and r = 3.
So, 5C5 = 4!/3!(4-3)! = (4 × 3 × 2 × 1)/(3 × 2 × 1) 1! = 24/6 = 4.
For combination when n = r, the number of combinations is always equal to 1.
See also: probability
