The number of seven digit positive integers formed using the digits 1,2,3 and 4 only and sum of the digits equal to 12 is________.

$$\begin{gathered} 5-1\text{'}\mathrm{s},1-4,1-3\to \frac{7!}{5!}=42 4-1\text{'}\mathrm{s},1-4,2-2\text{'}\mathrm{s}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\to \frac{7!}{4!2!}=105 \\ 4-1\text{'}\mathrm{s},1-2,2-3\text{'}\mathrm{s}\to \frac{7!}{4!2!}=105 3-1\text{'}\mathrm{s},3-2\text{'}\mathrm{s},1-3\text{'}\mathrm{s}\to \frac{7!}{3!3!}=140 \\ 2-1\text{'}\mathrm{s}5-2\text{'}\mathrm{s}\to \frac{7!}{2!5!}=21 ------413 \end{gathered}$$

or$$\begin{gathered} x_1+x_2+x_3+\dots x_7=12 \\ Number of solutions \\ \begin{array}{r}=\text{ Coefficient of }{x}^{12}\text{ in }{\left(x^1+x^2+x^3+x^4\right)}^7\\ =\text{ Coefficient of }{x}^5\text{ in }{\left(1+x+x^2+x^3\right)}^7\\ =\text{ Coefficient of }{x}^5\text{ in }{\left(1-x^4\right)}^7(1-x{)}^{-7}\\ =\text{ Coefficient of }{x}^5\text{ in }\left(1-7x^4\right)(1-x{)}^{-7}\\ =\text{ Coefficient of }{x}^5\text{ in }\left(1-7x^4\right)\sum _{r=0}^{\mathrm{\infty }}{ }^{7+r-1}{C}_r\cdot x^r\\ {=}^{11}{C}_5-7{\times }^7{C}_1\\ =462-49=413\end{array} \end{gathered}$$