Combinatorics

Question

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, ((⁷_k) (-x)^3k) * ((^7+k-1_k) x^k)

$((\binom{7}{k}) (-x)^{3k}) \times (\binom{7+k-1}{k} x^{k})$

Now not able to proceed. Kindly help.

Answer 1

I think it may help you

Comments

Thanks a lot.