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# Only 1% can solve this math problem

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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?

Apr 21, 2018

#1
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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.

Apr 21, 2018
#2
+101151
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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

Apr 21, 2018
#3
+1

CPhi, Congrats for solving this problem correctly. You are truly a genius

Guest Apr 22, 2018