<s>[INST] Number of distinct arrangement of $(a,b,c,d,e)$ 
If $a<b<c<d<e $ be


positive  integer such that


$a+b+c+d+e=20$.


Then number of distinct


arrangement of $(a,b,c,d,e)$ is

Here the largest value of $e$ is $10$
like $a\ b\ c\ d\ e$
as $ \ \  1\ 2\ 3\ 4\ 10$
And least value is $6$
like $ a\ b\ c\ d\ e$
as $\ \ 2\ 3\ 4\ 5\ 6$
Now after that solution given in book as
Total number of ways
$ \displaystyle =\binom{4}{0}+\frac{\binom{4}{1}}{4}+\frac{\binom{4}{2}}{3}+\frac{\binom{4}{3}}{2}+\frac{\binom{4}{4}}{1}$
$\displaystyle = 1+1+2+2+1=7$
I did not understand last $2$ line
i  e  solution given in book
Please have a look on that part
 [/INST] Equivalent problem; find non-negative integers $p\leq q\leq r\leq s\leq t$ such that $p+q+r+s+t=5$. The 5 numbers you want will be given by
$$
\begin{aligned}
a &= 1+p \\
b &= 2+q \\
c &= 3+r \\
d &= 4+s \\
e &= 5+t
\end{aligned}
$$
Repeated stars and bars and division by number of permutations will give you that expression
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