Distribution of identical balls in different boxes The number of ways in which $17$ identical red balls and $10$ identical white balls can be distributed among 4 distinct boxes such that number of red balls is greater then number of white balls in each box.
I am not able to think the logic . I thought of counting number of cases but the number is huge . Please help .
 A: The first step is to take equal number of white and red balls - $10$ each - and distribute them into $4$ boxes such that the number of white balls is same as the number of red balls in every box. In other words, you can take either $10$ white or $10$ red balls, distribute them in $4$ boxes and that fixes the distribution for the balls of the other color as it has to be same.
You can distribute $10$ balls in $4$ distinct boxes in $^{13}C_{10}$ ways using Stars and Bars method.
Now, the next step is to deal with the remaining $7$ red balls. After first step, number of white balls equals number of red balls in each box. As the number of red balls have to be more than the number of white balls in each box, place $1$ red ball in each of the $4$ boxes to ensure that. There is only $1$ way to do that as all the red balls are identical.
That leaves $3$ identical red balls. You can distribute them randomly to the $4$ boxes again applying Stars and Bars -
So, total number of ways = $^{13}C_{10} \times {^{6}C_{3}}$
