Our football team needs to fill 4 positions. To do this, it 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?
+983 Hello Guest!
Alright, first we need to take care of the positions with special conditions.
The lineman has a special condition, which only three of them are strong enough to do.
There are \(\binom{3}{1}\)ways to choose a lineman.
The rest doesn't need any special players.
So the total amount of ways is:
\(\binom{3}{1}+\binom{9}{1}+\binom{8}{1}+\binom{7}{1}=3+9+8+7=27\)
I'm pretty sure that is the final answer.
I hope this helps,
Gavin.
+129852 Here's my best attempt......
Assuming that only the ones who are strong enough to play offensive lineman only play that position...
We have 3 ways to choose these players for this position
And for the other 7, we need to choose any 3 of them to man the other positions
So..the total starting line-ups are
3 * C(7,3) = 3 * 35 = 105
However...if the the offensive lineman not selected for the offensive line can also play any of the other positions we have
3 ways to choose a lineman * 9 ways to fill any of the remaining 3 positions* 8 ways to choose any remaining two positions * 7 eays to choose the last position
C(3,1) * C(9,1) * C(8, 1) * C(7,1) = 3 * 9 * 8 * 7 = 1512 starting lineups
