CHAPTER 2 - Binomial Theorem for Positive Integral Index
Let us now consider more formally the binomial theorem. We need to expand , where , are two arbitrary quantities, but is a positive integer.
In particular, if we write
we need to find out the coefficient of . Note that must always equal , so that we can write a general term of the expansion (without the coefficient) as so that .
Now, to find the coefficient of , note that we need the quantity , times, while is needed times. Thus, in , will be formed whenever is ‘contributed’ by of the binomial terms, while is ‘contributed’ by the remaining of the binomial terms. For example, in the expansion of, to form , we need from terms and from :
How many ways are there to form ? In other words, how many times will be formed? The number of times is formed is what is the coefficient of . That number, which would be immediately obvious to the alert reader, is simply . Why? Because this is the number of ways in which we can select any binomial terms from . These terms will contribute . The remaining will automatically contribute .
In the general case of , we see that the coefficient of would be . (which is infact the same as ). Thus, the general binomial expansion is
The coefficients are called the binomial coefficients, for a reason that should now be obvious.
Note that the coefficient in this expansion is , which now explains the relation
we observed in the Pascal triangle; this relation simply corresponds to
Also, the binomial coefficients of terms equidistant from the beginning and the end are equal, because we have . The general term of expansion,, is the term from the beginning of the expansion and is conventionally denoted by , i.e.
Since we have terms in the general expansion, we see that if is even, there will be an odd number of terms, and thus there will be only one middle term, which would be . For example,
On the other hand, if is odd , then there will be an even number of terms in the expansion, and thus there will be two middle terms, namely and