How many solutions are there in the following equation over the natural numbers such that $x_1+x_2+x_3+x_4+x_5+x_6+x_7=30$ if $x_1+x_2+x_3>x_4+x_5+x_6+x_7$ ?
I made up a combinatorics question in my mind , but i stuck in it. What i tried is :
Firstly , i wanted to use symmetry property but , it has odd number of variables . Hence ,i could not do anything.
Secondly , i thought about whether generating functions can be used or not , but i stuck in writing the desired generating formula for it.
So , i hope to find nice approaches to my question.
$NOTE=$Natural numbers start with zero.
Best Answer
Using stars and bars you can write the number of solution as $$ \sum_{i=0}^{14}\binom{i+3}3\binom {30-i+2}2, $$ where the first factor counts the number of solutions to the equation $x_4+x_5+x_6+x_7=i $ and the second one those of $x_1+x_2+x_3=30-i $.