We have the following sets:
$X= {(a,b,c,d) ∈S: b< c < d},$
$Y= {(a,b,c,d) ∈S: a< c < d},$
$Z= {(a,b,c,d) ∈S: a< b < d},$
$F= {(a,b,c,d) ∈S: a< b < c},$
Where each of $a,b,c,d$ have integer values from 1 to 5
How to calculate $|X ∩ Y |$, $|X ∩ Z|$, $|Z ∩ F|$, $|X ∩ Y ∩ Z |$, $|X ∩ Y ∩ Z ∩ F|$ without the need to write down all possible combinations
I am sorry, but you will have to write down all possible combinations.
It's not too bad though.
For the first one, $a$ and $b$ are both less than $c$, which is less than $d$. So $c$ is at least 3, and at most 4. If $c=3$, there are two ways to do $a$ and $b$; for each of those there are two choices for 4. So that makes $2\times 2$ ways if $c=3$. If $c=4$, then $d$ must be $5$. There are three choices for $a$, and for each of those, there are two choices for $b$. So you work out how many ways with $c=4$.