Volleyball teams
Task number: 4489
Solution
The number of ways to divide 12 players into two unnamed teams of 6 is \(\tfrac12\binom{12}6\).
The number of ways to divide 11 players into two teams is \(\binom{11}{5}\), which can also be otherwise expressed as \(\binom{11}{6}\).
Since \(\binom{12}6=\binom{11}{5}+\binom{11}{6}\) holds for these binomial coefficients (see, e.g., Pascal triangle), we see that \(\tfrac12\binom{12}6=\binom{11}{5}\), or the number of possibilities does not change.
Even without calculating, the same result can be reached by reasoning that any 5+6 player distribution can be uniquely completed to a 6+6 distribution by adding Jirka to a five-player team, and vice versa, that leaving Jirka will yield a unique 5+6 distribution from every 6+6 distribution.
Answer
The number of possibilities will not change, in both cases there are \(\tfrac12\binom{12}6=\binom{11}{5}=462\) possibilities.
