  In the paper a simple explicit formula for an arbitrary $3j$-symbol for the
Lie algebra $\mathfrak{gl}_3$ is given. More precise necessary conditions for
non-vanishing of a $3j$-symbol are given, in the case when these conditions
hold we give an explicit expression for a $3j$-symbol. It is expressed through
a fraction of values of $A$-hypergeometric function when one substitutes $\pm
1$ instead of all it's arguments. The problem of calculation of an arbitrary
$3j$-symbol is equivalent to the problem of calculation of an arbitrary
Clebsh-Gordan coefficient for the algebra $\mathfrak{gl}_3$. These coefficients
play an important role in quantum mechanics in the theory of quarks.
