Problem 6

Let

p

be an odd prime number. How many

p

-element subsets

A

\{1,2,\ldots,2p\}

are there, the sum of whose elements is divisible by

p

Step 1 of 3: Remove the two exceptional subsets

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In plain words

The all-first and all-second blocks are already solutions and have fixed orbits.

\binom{2p}{p}-2

Detailed analysis

There are $\binom{2p}{p}$ total $p$ -element subsets. Exclude $\{1,\ldots,p\}$ and $\{p+1,\ldots,2p\}$ ; each has sum divisible by $p$ . Every remaining subset has $r$ elements in the first block with $0<r<p$ .