12 Combinations of 3

<-- 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 = 12 and r = 3, we get:
12C3 =12!
3!(12 - 3)!

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

Calculate the numerator n!:
n! = 12!
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
12! = 479,001,600

Calculate the first denominator (n - r)!:
(n - r)! = (12 - 3)!
(12 - 3)! = 9!
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
9! = 362,880

Calculate the second denominator r!:
r! = 3!
3! = 3 x 2 x 1
3! = 6

Now calculate our combination value nCr for n = 12 and r = 3:
12C3 =479,001,600
6 x 362,880

12C3 =479,001,600
2,177,280

12C3 = 220

In Microsoft Excel or , you write this function as =COMBIN(12,3)