# 52 Combinations of 5

<-- Enter Number of Items (n)
<-- Enter Number of Arrangements (r)

The formula for a combination of choosing r unique ways from n possibilities is:
 nCr = 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:
 52C5 = 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

Now calculate our combination value nCr for n = 52 and r = 5:
 52C5 = 80,658,175,170,943,876,845,634,591,553,351,679,477,960,544,579,306,048,386,139,594,686,464 120 x 258,623,241,511,168,177,673,491,006,652,997,026,552,325,199,826,237,836,492,800

 52C5 = 80,658,175,170,943,876,845,634,591,553,351,679,477,960,544,579,306,048,386,139,594,686,464 31,034,788,981,340,182,748,066,613,504,319,524,244,564,993,428,643,676,761,882,624

52C5 = 2,598,960

In Microsoft Excel or , you write this function as =COMBIN(52,5)