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Splitting into $2$ teams

CombinatoricsMedium

$10$ distinct people are split into $2$ teams of $5$ for a case competition. The teams are not labelled, so swapping the teams gives the same split. How many different splits exist?

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Hint

Count labelled splits first, then correct for the swap.

Solution

Choosing the first labelled team gives $(5 10) = 252$ options.
Each unlabelled split is counted $2$ times, once per labelling.
Divide: $252 \div 2$ .
Therefore the number of splits is $126$ .

Answer: $126$

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