19 Combinations of 2

<-- 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 = 19 and r = 2, we get:
19C2 =19!
2!(19 - 2)!

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

Calculate the numerator n!:
n! = 19!
19! = 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
19! = 121,645,100,408,832,000

Calculate the first denominator (n - r)!:
(n - r)! = (19 - 2)!
(19 - 2)! = 17!
17! = 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
17! = 355,687,428,096,000

Calculate the second denominator r!:
r! = 2!
2! = 2 x 1
2! = 2

Now calculate our combination value nCr for n = 19 and r = 2:
19C2 =121,645,100,408,832,000
2 x 355,687,428,096,000

19C2 =121,645,100,408,832,000
711,374,856,192,000

19C2 = 171

In Microsoft Excel or , you write this function as =COMBIN(19,2)