15 Combinations of 10

<-- 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 = 15 and r = 10, we get:
15C10 =15!
10!(15 - 10)!

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

Calculate the numerator n!:
n! = 15!
15! = 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
15! = 1,307,674,368,000

Calculate the first denominator (n - r)!:
(n - r)! = (15 - 10)!
(15 - 10)! = 5!
5! = 5 x 4 x 3 x 2 x 1
5! = 120

Calculate the second denominator r!:
r! = 10!
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
10! = 3,628,800

Now calculate our combination value nCr for n = 15 and r = 10:
15C10 =1,307,674,368,000
3,628,800 x 120

15C10 =1,307,674,368,000
435,456,000

15C10 = 3,003

In Microsoft Excel or , you write this function as =COMBIN(15,10)