In how many ways can a team be formed when some players can play multiple positions? I want to form a lineup where we have 1 goalkeeper, 2 defenders and 3 forwards. We have 3 goalkeepers, 7 defenders and 10 forwards available, as well as 4 players who can play as defender or forward. How many possible lineups are there?
My approach was to separately consider the cases where $n$ of the 4 players (lets call them hybrid players) who can play as defender or forward are first chosen as defenders or forwards. If we first pick the 2 defenders, we can pick 0, 1 or 2 of the hybrid players as defenders, and then either 4, 3 or 2 of these players will be left available for forward. I tried calculating each case separately, and summing the combinations for each case together. However, if I calculate it such that we first choose the forwards, and we can choose 3, 2, 1 or 0 of the hybrid players to be forwards, I get a different result. I assume there is a smarter way to do this, but I can't seem to figure it out.
 A: Assuming you consider the positions within each position group irrelevant, you would still want to consider, for a given selection of hybrids, which position group each of the hybrids play in. Further, given that the GK selection is independent of the rest of the selections, that is simply a factor of 3 on the final answer, so I'll ignore it from here. The solution is then broken up based on how many hybrids are selected, where the hybrids play, and any remaining selections. I will include a sample term with reasoning, and the remaining terms are left as an exercise.
Consider the case where 3 hybrids are selected , two of them playing as forwards, one as a defender. Then the term is:
$${4 \choose 3} \cdot {3 \choose 1} \cdot {10 \choose 1} \cdot {7 \choose 1}$$
With each factor corresponding respectively to combinations of:

*

*hybrid selections

*hybrid positions

*forward selections

*defender selections

By iterating over the number of selected hybrids and the possible position combinations they play, I reach 16972 combinations before considering the GK.
A: Partitioning the $4$ hybrid players into $d$ defenders, $f$ forwards, and $4-d-f$ unused players yields
$$
\binom{3}{1}\sum_{d=0}^2\sum_{f=0}^{\min(4-d,3)}\binom{4}{d,f,4-d-f}\binom{7}{2-d}\binom{10}{3-f} = 50916
$$
