Show that $\displaystyle 1+ \sum\limits_{k=1}^{n} k \cdot k! = (n+1)!$
RHS: This is the number of permutations of an $n+1$ element set. We can rewrite this as $n!(n+1)$.
LHS: It seems that the $k \cdot k!$ has a similar form to $(n+1)! = (n+1)n!$ Also we can write $1 = 0!$ I think you use the mulitplication principle is being used here (e.g. the permutations of a $k$ element set multiplied by $k$).
Note that a combinatorial proof is wanted (not an algebraic one).