The number of seven digit positive integers formed using the digits 1,2,3 and 4 only and s…

Question

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 \_\_\_\_\_\_\_\_\_\_\_.

Answer: 413

Accepted Answer

${x}_{1}+{x}_{2}+{x}_{3}+\ldots {x}_{7}=12$. This equation represents the number of ways to distribute 12 identical items (the sum of the digits) into 7 distinct boxes (the seven digits of the number), where each box can contain one of the numbers 1, 2, 3, or 4.

Number of solutions

$\begin{gathered} ={x}^{12}{({x}^{1}+{x}^{2}+{x}^{3}+{x}^{4})}^{7} \ ={x}^{5}{(1+x+{x}^{2}+{x}^{3})}^{7} \ ={x}^{5}{(1-{x}^{4})}^{7}(1-x{)}^{-7} \ ={x}^{5}(1-7{x}^{4})(1-x{)}^{-7} \ ={x}^{5}(1-7{x}^{4})\sum _{r=0}^{\infty}{}^{7+r-1}{C}_{r}\cdot {x}^{r} \ ={}^{11}{C}_{5}-7	imes {}^{7}{C}_{1} \ =462-49=413 \end{gathered}$

Solutions · 1