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1.23 Binomial Theorem

{{n}\choose{k}}=(k^{th}\quad{\hbox{entry in row}}\quad n)

where the entries in the array are given by the recurrence relation

{{n}\choose{k}}+{{n}\choose{k+1}}={{n+1}\choose{k+1}}.

Theorem

Let $n\,$ be a positive integer. Then

(1+X)^{n}=1+{{n}\choose{1}}X+{{n}\choose{2}}X^{2}+\dots+{{n}\choose{n}}X^{n},

P_{n}

where the binomial coefficients are given by the $n^{th}$ row of Pascal’s triangle.