Okay. Took me while.
Basically you arrange the coins in order, quarters, loonies, and toonies and you make a check for each coin. When a coin switches value you put a divider.
So example if you have QQLT, you write $\checkmark\checkmark|\checkmark|\checkmark$ meaning two quarters (check, check) we switch to loonies (divider), one loony (check), we switch to toonies (divider), a final toonie (check).
Likewise for QTTT, you write $\checkmark||\checkmark\checkmark\checkmark$. That is one loonie (check), switc to loonie (divide), switch to toonie (divide), three toonies (check, check, check) and $LLLL$ would be $|\checkmark\checkmark\checkmark\checkmark|$ etc.
So how many ways are there to do this. You have six things. $a=\checkmark,b=\checkmark,c=\checkmark,d=\checkmark,e=|,f=|$ which may be checks or dividers. There are $6!= 720$ ways to put six things in order. But of those $720$ ways, whatever order the four checkmarks are, are considered the same. That is QLLT = $a\checkmark, e|, b\checkmark, c\checkmark, f|,d\checkmark$ is the exact same thing as QLLT $d\checkmark, f, a\checkmark,b\checkmark, e|, c\checkmark$
So there are $4!= 24$ ways to plays the checkmarks that are considered exactly the samme and $2! = 2$ ways to place the dividers that are considered exactly the same.
So the total way to do the if $\frac {6!}{4!2!} = \frac {1*2*3*4*5*6}{(1*2*3*4)*(1*2)} = \frac {5*6}{1*2} = 15$.
That are:
QQQQ = $\checkmark\checkmark\checkmark\checkmark||$
QQQL = $\checkmark\checkmark\checkmark|\checkmark|$
QQLL = $\checkmark\checkmark|\checkmark\checkmark|$
QLLL = $\checkmark|\checkmark\checkmark\checkmark|$
LLLL = $|\checkmark\checkmark\checkmark\checkmark|$
QQQT = $\checkmark\checkmark\checkmark||\checkmark$
QQLT = $\checkmark\checkmark|\checkmark|\checkmark$
QLLT = $\checkmark|\checkmark\checkmark|\checkmark$
LLLT = $|\checkmark\checkmark\checkmark|\checkmark$
QQTT =$\checkmark\checkmark||\checkmark\checkmark$
QLTT = $\checkmark|\checkmark|\checkmark\checkmark$
LLTT = $|\checkmark\checkmark|\checkmark\checkmark$
QTTT = $\checkmark||\checkmark\checkmark\checkmark$
LTTT = $|\checkmark|\checkmark\checkmark\checkmark$
TTTT = $||\checkmark\checkmark\checkmark\checkmark$