The binomial theorem gives the coefficients of the expansion of powers of binomial expressions. A binomial expression is simply the sum of two terms, such as x + y.
Example. The expansion of 
can be found using combinatorial reasoning instead of multiplying the three terms out. When 
is expanded, all products of a term in the first sum, a term in the second sum, and a term in the third sum added. Terms of the form x 3, x 2 y, xy 2, and y 3 arise. To obtain a term of the form x 3, an x must be chosen in each of the sums, and this can be done in only one way. Thus, the x 3 term in the product has a coefficient of 1. To obtain a term of the form x 2 y, an x must be chosen in two of the three sums (and consequently a y in the other sum). Hence, the number of such terms is the number of 2-combinations of three objects, namely C (3, 2). Similarly, the number of terms of the form xy 2 is the number of ways to pick one of the three sums to obtain an x (and consequently take a y from each of other two terms). This can be done in C (3, 1) ways. Finally, the only way to obtain a y 3 term is to choose the y for each of the three sums in the product, and this can be done in exactly one way. Consequently, it follows that 
Theorem 3 (the binomial theorem). Let x and y be variables, and let n be a positive integer. Then
Proof: The terms in the product when it is expanded are of the form 
for 
To count the number of terms of the form , note that to obtain such a term it is necessary to choose n – j x s from the n sums (so that the other j terms in the product are y s). Therefore, the coefficient of is 
Example. What is the expansion of 
?
Solution: 
.
Example. What is the coefficient of 
in the expansion of 
?
Solution: 
. Consequently, the coefficient of in the expansion is obtained when j = 13, namely, 
.
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