  Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra
over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1,
x_2, \ldots , x_s$, each of degree one. This algebra is considered as a graded
module over the mod-2 Steenrod algebra, $\mathscr {A}$. We are interested in
the "hit" problem of finding a minimal set of generators for $\mathscr
A$-module $P_s.$ This problem is unresolved for every $s\geqslant 5.$ In this
paper, we study the hit problem of five variables in a generic degree, from
which we investigate Singer's conjecture [Math. Z. 202 (1989), 493-523] for the
transfer homomorphism of rank $5$ in degrees given. This gives an efficient
method to study the algebraic transfer and it is different from the ones of
Singer.
