Pascal Triangle

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In Probability,

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

{n-1 \choose k-1} + {n-1 \choose k}

= {(n-1)! \over {(k-1)!~(n-k)!}} + {(n-1)! \over {(n-1-k)!~k!}}

= ({k \over k} {(n-1)! \over {(k-1)!~(n-k)!}}) + ({(n-1)! \over {(n-1-k)!~k!}} {(n-k) \over (n-k)})

= {{{(n-1)!} (k + n - k)} \over {k!~(n-k)!}} = {n! \over {k!~(n-k)!}}

= {n \choose k} ~~~~~~~~~~ \blacksquare

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