The Binomial Expansion
In Chapter 3, we introduced a family of numbers which were denoted by 
Given integers n and r with 
the number 
is defined as the number of r-element subsets of the set 
n = {1,2,..., n}. That is, 
is the number of ways of selecting r distinct objects from a set of n distinct objects. We also derived the following formula for :
By applying (10.1), or otherwise, we can easily derive some interesting identities involving these numbers such as:
In this chapter, we shall learn more about this family of numbers and derive some other important identities involving them.
In algebra, we learn how to expand the algebraic expression (1 + x)n for n = 0,1, 2, 3. Their expansions are shown below:
Notice that the coefficients in the above expansions are actually numbers of the form . Indeed, we have:
What can we say about the coefficients in the expansion of (1 + x)4? Will we obtain
Let us try to find out the coefficient of x2 in the expansion of (1+x)4. We may write
Observe that in the expansion, each of the factors (1), (2), (3) and (4) contributes either 1 or x, and they are multiplied together to form a term. For instance, to obtain x2 in the expansion, two of (1), (2), (3) and (4) contribute x and the remaining two contribute 1. How many ways can this be done? Table 10.1 shows all the possible ways, and the answer is 6.
Thus, there are 6 terms of x2 and the coefficient of x2 in the expansion of (1 + x)4 is therefore 6. Indeed, to select two x’s from four factors (1 + x), there are 
ways (while the remaining two have no choice but to contribute “1”). Thus the coefficient of x2 in the expansion of (1 + x)4 is which is 6. Using a similar argument, one can readily see that
In general, what can be said about the expansion of (1 + x)n, where n is any natural number?
Let us write
To expand (1 + x)n, we first select 1 or x from each of the n factors (1 + x), and then multiply the n chosen 1’s and x’s together. The general term thus obtained is of the form xr, where 0 ≤ r ≤ n. What is the coefficient of xr in the expansion of (1 + x)n if the like terms are grouped? This coefficient is the number of ways to form the term xr in the product (*). To form a term xr, we choose r factors (1 + x) from the n factors (1 + x) in (*) and select x from each of the r chosen factors. Each of the remaining n – r factors (1 + x) has no other option but to contribute 1. Clearly, the above selection can be done in ways. Thus, the coefficient of xr in the expansion of (1 + x)n is given by . We thus arrive at the following result:
Exercise
10.1 By applying Identity (10.1), or otherwise, derive the following identities:
(i) 
(ii) 
10.2 In the expansion of (1 + x)100, it is known that the coefficients of xr and x3r, where 1 ≤ r ≤ 33, are equal. Find the value of r.
10.3 What is the largest value of k such that there is a binomial expansion (1 + x)n in which the coefficients of k consecutive terms are in the ratio 1:2:3: … : k? Identify the corresponding expansion and the terms.
10.4 Find the terms in the expansion of (1 + 3x)23 which have the largest coefficient.
