Problem:
Our football team has 10 members, of which only 3 are strong enough to play offensive lineman, while all other positions can be played by anyone. In how many ways can we choose a starting lineup consisting of a quarterback, a running back, an offensive lineman, and a wide receiver?
Source: AoPS Alcumus
My Method:
I first tried the obvious $^3C_1\cdot^9C_3 = 3\cdot84=252$ i.e. choose a lineman from the available three and then the other three team members from the nine left.
But, my answer was wrong and the correct method was to do $3\cdot9\cdot8\cdot7=1512$ which also seems correct, as there are 3 choices for the offensive lineman position, then there are 9 choices for the next position, 8 choices for the position after, and 7 choices for the last position. Thus, my answer was less by 6 times.
Question:
- How do I decide which method to use?
- Where did I undercount in my method? It does look well?
Your answer assumes that the lineman is distinguishable from the other positions, but that the other three positions are indistinguishable from each other. So, it solves the problem of choosing a lineman and three other players. But these three other players can choose their positions in $3!=6$ ways. Their answer is naturally based off of permutations and directly gets the correct answer.