Binominal Coefficients

 

INTRODUCTION

 

If we have the sum of two quantities raised to some power, we can immediately write the resulting polynomial without having to perform all the intermediate multiplications using the expansion as follows:

 

Or for example if n=5, we might write

 

 

In the above expression, the “binominal coefficients” are the terms.  These are the simple combinatorials of “n choose j” or the number of ways we can select “j” items from a total set of “n” uniquely identifiable items.  These binominal coefficients are given by

 

 

For this equation, please note that both the numerator and the denominator always have “j” terms.   And so for our example above for n=5, we have

 

 

 

 

 

 

And for j=0, we always define the combinatorial

 

 

PASCAL’S TRIANGLE

 

As an interesting side note, binominal coefficients can be easily calculated from a series of sums.  To do this we construct “Pascal’s Triangle” in which each term is the sum of the two terms above it in the preceding row.  For the first six rows we have

 

 

Note that the sixth row is simply an array of the combinatorials indicated.  And please note as well, that the next line would describe the combinatorials  for j = 0, 1, …, 6.