1. ## Factorial Problem

Hi I have a problem here is the question:

Ten students have been nominated for positions of secretary, social convenor, treasurer, fundraising chair. In how many ways can these positions be filled if the Norman twins are running and plan to switch positions on occasion for fun since no one can tell them apart?

Thanks for having me here on Math Help Forum as this is my first post

2. Originally Posted by jassy316
Hi I have a problem here is the question:

Ten students have been nominated for positions of secretary, social convenor, treasurer, fundraising chair. In how many ways can these positions be filled if the Norman twins are running and plan to switch positions on occasion for fun since no one can tell them apart?

Thanks for having me here on Math Help Forum as this is my first post
Hi Jassy316,

If only one Norman twin was running for this,
there would be 9 from which to choose 4.

If both Norman twins are running and they are distinguishable,
(different clothes, different hairstyles etc),
then there are 10 available from which to choose 4.

Therefore the 2nd twin introduces an extra $\displaystyle 10p_4-9p_4$ possible arrangements.

If they decide to be indistinguishable, then when 1 of them is in the group,
the other can swop places undetected.
This makes them effectively a group of 2 indistinguishable members, for all the arrangements either one is in.

When they are together, 2 of the 4, indistinguishable, they are effectively an identical group of 2.

Hence we divide the added arrangements contributed by the 2nd twin by 2!

Therefore, the number of arrangements is

$\displaystyle 9p_4+\frac{10p_4-9p_4}{2!}$