We show that general $3n-j (n>2)$ symbols of the first kind and the second
kind for the group SU(2) can be reformulated in terms of binomial coefficients.
The proof is based on the graphical technique established by Yutsis, et al. and
through a definition of a reduced $6-j$ symbol. The resulting $3n-j$ symbols
thereby take a combinatorial form which is simply the product of two factors.
The one is an integer or polynomial which is the single sum over the products
of reduced $6-j$ symbols. They are in the form of summing over the products of
binomial coefficients. The other is a multiplication of all the triangle
relations appearing in the symbols, which can also be rewritten using binomial
coefficients. The new formulation indicates that the intrinsic structure for
the general recoupling coefficients is much nicer and simpler, which might
serves as a bridge for the study with other fields. Along with our newly
developed algorithms, this also provides a basis for a direct, exact and
efficient calculation or tabulation of all the $3n-j$ symbols of the SU(2)
group for all range of quantum angular momentum arguments. As an illustration,
we present teh results for the $12-j$ symbols of the first kind.