A combination is a way to order or arrange a set or number of things (uniquely)

The formula for a combination of choosing runique ways from n possibilities is:

_{n}C_{r} =

n!

r!(n - r)!

where n is the number of items and r is the unique arrangements.

Plugging in our numbers of n = 52 and r = 5, we get

_{52}C_{5} =

52!

5!(52 - 5)!

Remember from our factorial lesson that n! = n * (n - 1) * (n - 2) * .... * 2 * 1

Calculate the numerator n!:

n! = 52! 52! = 52 x 51 x 50 x 49 x 48 x 47 x 46 x 45 x 44 x 43 x 42 x 41 x 40 x 39 x 38 x 37 x 36 x 35 x 34 x 33 x 32 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 52! = 80,658,175,170,943,876,845,634,591,553,351,679,477,960,544,579,306,048,386,139,594,686,464

Calculate the first denominator (n - r)!:

(n - r)! = (52 - 5)! (52 - 5)! = 47! 47! = 47 x 46 x 45 x 44 x 43 x 42 x 41 x 40 x 39 x 38 x 37 x 36 x 35 x 34 x 33 x 32 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 47! = 258,623,241,511,168,177,673,491,006,652,997,026,552,325,199,826,237,836,492,800

Calculate the second denominator r!:

r! = 5! 5! = 5 x 4 x 3 x 2 x 1 5! = 120

Calculate our combination value _{n}C_{r} for n = 52 and r = 5

How does the Permutations and Combinations Calculator work?

Calculates the following: Number of permutation(s) of n items arranged in r ways = _{n}P_{r} Number of combination(s) of n items arranged in r unique ways = _{n}C_{r} including subsets of sets This calculator has 2 inputs.

What 2 formulas are used for the Permutations and Combinations Calculator?

What 4 concepts are covered in the Permutations and Combinations Calculator?

combination

a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter _{n}P_{r} = n!/r!(n - r)!

factorial

The product of an integer and all the integers below it

permutation

a way in which a set or number of things can be ordered or arranged. _{n}P_{r} = n!/(n - r)!

permutations and combinations

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