Question

6. Five freshmen and five sophomores stand in a line. How many arrangements are possible if
a. there are no restrictions?
b. if all of the freshmen stand next to each other?
c. if the freshmen and sophomores alternate?

Answer 1

this is a basic combinatorics problem.
in the first case, you have no restrictions
so you can choose any of the ten in the first place, then any of the remaining nine for the second, then any of the remaining eight for the third, etc
so you get 10*9*8*7*6*5*4*3*2*1
the mathematical way of writing this is 10! so 10!=10*9*8*7...3*2*1

the second case is a bit tricky but still simple to solve. if all the freshies are together than you basically have five different combinations of how many sophomores before and after the freshies in addition to the combinations of which freshman goes first, which second etc.

if you look at them this way, this is your combinations of fresh vs soph (not considering individual yet, just group):

FFFFFSSSSS
SFFFFFSSSS
SSFFFFFSSS
SSSFFFFFSS
SSSSFFFFFS
SSSSSFFFFF

so we have 6 combinations of sophs vs fresh, lets look at the combinatorics of the orders with reference to each:

FFFFFSSSSS = 5! * 5!
SFFFFFSSSS = 5 * 5! * 4!
SSFFFFFSSS = 5*4*5!*3!
SSSFFFFFSS = 5*4*3*5!*2!
SSSSFFFFFS = 5*4*3*2*5!*1
SSSSSFFFFF = 5!*5!

you should be able to see why i got those equations, but if not then just ask for clarification.
if you look at those six cases, youll notice that they all equal 5!*5! or 14400.
so we can then say that we have six cases of 14,400 combinations so we have a ottal of 6*14,400 combinations or 86,400 combinations.

the last one is almost as easy as the first:

you have either
FSFSFSFSFS
or
SFSFSFSFSF

and in both cases you basically have
"choose any of the five S, now choose any of the five F, now choose any of the remaining four S, now any of the remaining four F... etc etc"
or you have
"choose any of the five F, now choose any of the five S, now choose any of the remaining four F, now any of the remaining four S... etc etc"

both give you 5*5*4*4*3*3*2*2*1*1 which is the same as 5*4*3*2*1*5*4*3*2*1 which is the same as 5!*5! which we know is 14400.
since we have two options of order, that is, starting with freshies vs sophs, we just multiply that by two and end up with 14400*2 or 28800 combinations