## Computing Binomial Coefficients C(n, r)

On the right is a bag of 10 marbles with 7 of them visible.
Q How many possible sets of 7 marbles are there?
A The number of possible sets of 7 marbles is called C(10, 7). Other notations for it are10C7 and
 ( 107 )
.
We can compute C(m, n) either by hand or using technology. Here are some examples to illustrate by-hand computation.

Computing C(10, 7) by hand

1. Create a faction with 10 in the numerator and 7 in the denominator:  107
2. Extend the term in the denominator to its factorial:  107.6.5.4.3.2.1
3. Extend the term in the denominator to as many terms as in the denominator:  10.9.8.7.6.5.4  7.6.5.4.3.2.1
4. Compute it (it helps to first cancel the terms in the denominator):  10.9.8.7.6.5.4  7.6.5.4.3.2.1
=120
Thus, C(10, 7) = 120.
More Examples
C(5, 2) =  5.42.1
=10 C(5, 5) =  5.4.3.2.15.4.3.2.1
=1

Some Special Cases:

C(n, 0) = 1               C(n, n) = 1

Computation Utility:

To compute C(n, r), j ust enter n and r and press "Compute". The answer will appear after the equals sign.
 C( , ) =