Is there a way to show X equal or smaller than Y?

Question

X=$1!+2!+3!+...+n!$

Y=$n+n(n-1)+n(n-1)(n-2)+....+n(n-1)(n-2)(n-3)...(n-a)$

Is there a way to show X equal or smaller than Y?

for $a= n-1$ and and $a=n-2$

i found that

Y=$n+n(n-1)+n(n-1)(n-2)+..+n!+n!$

However after this point do not find a logical connection and do not have idea about solving this problem.

Answer 1

Without additional hypotheses (e.g., some lower limit on $a$) the statement is false in general. For example, if $n=4$ and $a=1$ then $X=31$ but $Y=16.$