Adding up every multinomial coefficient of order 7

Consider every ordered quadruple $(k_1,k_2,k_3,k_4)$ of non-negative integers summing to $7$. Find the total of the multinomial coefficients $\binom{7}{k_1,\,k_2,\,k_3,\,k_4}$ over all of them.

$16384$. Each coefficient counts ways to put $7$ labelled items into $4$ labelled boxes with the given box sizes, so the total counts all assignments: $4^7$. Equivalently, set every variable to $1$ in $(x_1+x_2+x_3+x_4)^7$.