Binominal Theorem

The binomial theorem expresses the expansion of \({ (a+b)} ^{ n} \), where a and b are terms and n is a non-negative integer.

The formula is:

\({ (a+b)} ^{ n} = \sum_{ k=0} ^{ n} (\frac{ n} { k} ){ a} ^{ n-k} { b} ^{ k} \)

where \((\frac{ n} { k} )\) (binomial coefficient) is defined as:

\((\frac{ n} { k} ) = \frac{ n!} { k!(n-k)!} \)

These coefficients represent the number of ways to choose k items from n items.

They can be found in Pascal's triangle, where each entry is the sum of the two directly above it.

The sum of the coefficients of the expansion of \({ (a+b)} ^{ n} \) is \({ 2} ^{ n} \) (setting a = b = 1).

One of many applications is expanding powers of binomials in algebraic manipulations and simplifications.

For example, calculating \({ (x+y)} ^{ 5} \) is possible without direct multiplication.