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Binomial Coefficient

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The binomial coefficients are numbers that can be used to count the number of choosing ‘k’ objects from a collection of ‘n’ objects.

The binomial coefficients are denoted by the symbol nCk or n \choose k (pronounced as “n choose k”). We can calculate the value of the binomial coefficient by using the formula,

{n \choose k} = \frac{n!}{k!(n-k)!}

where, n! = n.(n-1)\ldots 3.2.1 that is, the product of all integers from 1 to n.

The above formula tells us the number of ways in which we can select ‘k’ objects out of ‘n’. For example, if we want to find the number of ways of choosing 3 people out of a group of 8, we calculate the binomial coefficient 8 \choose 3 (pronounced as “8 choose 3”).

Example 1:

We calculate the binomial coefficient 8 \choose 3 as follows:

{8 \choose 3} = \frac{8!}{3!5!} = \frac{8*7*6*5*4*3*2*1}{3*2*1 \text{ x } 5*4*3*2*1}

The factors 5, 4, 3, 2, 1 get cancelled in both the numerator and denominator and we get,

{8 \choose 3} = \frac{8*7*6}{3*2*1} = 56

Example 2:

We calculate the binomial coefficient 4 \choose 2 as follows:

{4 \choose 2} = \frac{4!}{2!2!} = \frac{4*3*2*1}{2*1 \text{ x } 2*1} {4 \choose 2} = \frac{4*3}{2*1} = 6

Remark: 1) The reason behind these numbers being called binomial coefficients is that they occur in the formula for the binomial expansion.

2) We can calculate the values of the binomial coefficients on a TI-84 calculator. To calculate {4 \choose 2} we first type 4, then click on MATH, go to PRB and choose the option nCr and then type 2 and hit ENTER.

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