Find the number of seven digit integers with sum of the digits equal to $11$ and formed by using the digits $1,2$ and $3$ only.
Soln- $X_{1}+X_{2}+.......X_{7}=11$
$(x+x^{2}+x^{3})^{7}$
$(x(1+x+x^{2}))^{7}$
$x^{7}(1+x+x^{2})^{7}$
$x^{7}(\dfrac{1-x^{3}}{1-x})^{7}$
To find coefficient of $X^{11}$ we have to find coefficient of $x^{4} $in $(\frac{1-x^{3}}{1-x})^{7}$
so, ((7k) (-x)3k) * ((7+k-1k) xk)
$((\binom{7}{k}) (-x)^{3k}) \times (\binom{7+k-1}{k} x^{k})$
Now not able to proceed. Kindly help.