There are 12 students interested in creating a shoe shining club. There will be a president, a vice-president, 2 shoe shiners, 2 polish selectors, and the rest will work on advertising.

Question

There are 12 students interested in creating a shoe shining club.  There will be a president, a vice-president, 2 shoe shiners, 2 polish selectors, and the rest will work on advertising.
How many ways can these positions be filled?

I did: (12C1)+(12C1)+(12C2)+(12C2)+(12C6) and got 1015 ways as my answer. Is this correct?

Reiny · Accepted Answer

No!

The president can be chosen in C(12,1) ways
the VP can be chosen in C(11,1) ways
the 2 shoe-shiners in C(10,2) ways
the 2 polish selectors in C(8,2) ways, leaving C(6,6) for advertising.

multiply these ....
12x11x45x28x1 = 166320

Anonymous · Answer

Thank you Reiny! So just for clarification, you reduce n by how many were chosen previously because those people were already selected and you can't chose them anymore, right?