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Can some one please show me how to prove directly that $S(n, n - 1) = \binom n 2$ I know that $S(n,n)=1$

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Your question is not clear (Make sure you are using proper notations) –  Kirthi Raman Feb 22 '12 at 17:09

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If you want to partition $n$ objects into $n-1$ nonempty sets, then exactly one of the sets will have two elements and the rest will all have one. It is thus equivalent to decide which two elements will be paired together, which can be done in $\binom{n}{2}$ ways.

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