Calculating Binomial Coefficients

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 are

₁₀C₇

and

(

10
7

)

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:

2. Extend the term in the denominator to its factorial:

7^.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

More Examples

C(5, 2) =

5^.4

2^.1

C(5, 5) =

5^.4^.3^.2^.1

Some Special Cases:

n, n

Computation Utility:

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