QJEEM23Ma8730: JEE Main 2023 Mathematics - Class 11 JEEM23Ma8730 Solution: The number of seven digit positive integers formed using the…

$x_1+x_2+x_3+\ldots x_7=12$ . 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

$\begin{aligned}& =\text { Coefficient of } x^{12} \text { in }\left(x^1+x^2+x^3+x^4\right)^7 \\\\& =\text { Coefficient of } x^5 \text { in }\left(1+x+x^2+x^3\right)^7 \\\\& =\text { Coefficient of } x^5 \text { in }\left(1-x^4\right)^7(1-x)^{-7} \\\\& =\text { Coefficient of } x^5 \text { in }\left(1-7 x^4\right)(1-x)^{-7} \\\\& =\text { Coefficient of } x^5 \text { in }\left(1-7 x^4\right) \sum_{r=0}^{\infty}{ }^{7+r-1} C_r \cdot x^r \\\\& ={ }^{11} C_5-7 \times{ }^7 C_1 \\\\& =462-49=413\end{aligned}$