2.6: The Binomial Theorem
Here is a truly basic result from combinatorics kindergarten.
Let \( x\) and \( y\) be real numbers with \( x\), \( y\) and \( x+y\) non-zero. Then for every non-negative integer \(n\) ,
\((x+y)^n = \displaystyle \sum_{i=0}^n \dbinom{n}{i} x^{n-i}y^i\)
- Proof
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View \((x+y)^n\) as a product
\((x+y)^n = \underbrace{(x+y)(x+y)(x+y)(x+y)...(x+y)(x+y)}_{n factors}\).
Each term of the expansion of the product results from choosing either \(x\) or \(y\) from one of these factors. If \(x\) is chosen \(n−i\) times and \(y\) is chosen \(i\) times, then the resulting product is \(x^{n-i}y^i\). Clearly, the number of such terms is \(C(n,i)\), i.e., out of the \(n\) factors, we choose the element \(y\) from \(i\) of them, while we take \(x\) in the remaining \(n-i\).
There are times when we are interested not in the full expansion of a power of a binomial, but just the coefficient on one of the terms. The Binomial Theorem gives that the coefficient of \(x^5y^8\) in \((2x-3y)^{13}\) is \(\binom{13}{5}2^5(-3)^8\).
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