A box contains $7$ distinct red marbles, $6$ distinct green marbles and $5$ distinct yellow marbles. How many ways are there to get $8$ balls of all $3$ colors, at least one of each color and that order doesn't matter ?
My attempt : This is my generating function : Sage:
$f= (x+x^2/2+x^3/6+x^4/24+x^5/120)*(x+x^2/2+x^3/6+x^4/24+x^5/120+x^6/720)*(x+x^2/2+x^3/6+x^4/24+x^5/120+x^6/720+x^7/7!)$
My answer: $5740$. Unfortunately, this is the wrong answer.
Can someone help me to find the correct answer? Thank in advance.
Since each red marble is said to be distinct, same with green and yellow, you need to find the coefficient of $x^8$ in the expression
$[(7 x + 21 x^2 + 35 x^3 + 35 x^4 + 21 x^5 + 7 x^6 + x^7) (6 x + 15 x^2 + 20 x^3 + 15 x^4 + 6 x^5 + x^6) (5 x + 10 x^2 + 10 x^3 + 5 x^4 + x^5)]$
which comes to be $41811$
[ You have to choose the marbles from each color in the function, and use an $OGF$ ]