# Number of solutions of $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 34$ in positive even integers not exceeding 10

"Find the number of solutions of $$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 34$$ in positive even integers not exceeding 10."

Answer: $$\binom{16}{5} - 6\binom{11}{5} + 15\binom{6}{5}$$

According to the book "Maths of Choice, the answer was derive from the following: "the question amounts to asking for the number of solutions of $$y_1 + y_2 + ... + y_6 = 17$$ in positive integers not exceeding 5, because any even integer $$x_1$$ can be written as $$2y_1$$, where $$y_1$$ is again an integer.

Below is how I think the solution is derived; however, I'm not sure where the 5s from $$\binom{17-5-1}{6-1}$$ and $$\binom{17-5-5-1}{6-1}$$ are from. Are they the positive even integers between 1 to 10 or "17 in positive integers not exceeding 5"?

n: $$\binom{17-1}{6-1} = \binom{16}{5}$$

n(a) = $$\binom{17-5-1}{6-1} = \binom{11}{5}$$

n(ab) = $$\binom{17-5-5-1}{6-1} = \binom{6}{5}$$

• The author first transformed the problem of finding the number of solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 34$ in positive even integers not exceeding $10$ to finding the number of solutions to the equation $y_1 + y_2 + y_3 + y_4 + y_5 + y_6 = 17$ in positive integers not exceeding $5$. Therefore, the author is subtracting cases in which one or more of the variables in the second equation I wrote exceeds $5$. May 23, 2022 at 8:54