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Suppose you pick a k-element subset of {1, 2, ..., n}, call it A. How many of the other k-element subsets have k-1 elements in common with A?

I've been at this for quite some time, but I always overcount. Can anyone help me out? My last attempt gave me [itex](n-k+1)k - \frac{k(k-1)}{2}[/itex], which isn't always correct. I feel I'm missing something simple.

I've been at this for quite some time, but I always overcount. Can anyone help me out? My last attempt gave me [itex](n-k+1)k - \frac{k(k-1)}{2}[/itex], which isn't always correct. I feel I'm missing something simple.

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