$50$ prizes to be distribute among $4$ teams $A$ ,$B$, $C$ ,$D$ , provided that the team $A$ gets $15$ prizes at most ; and team B gets $20$ prizes at most, How many ways could we do that ?
My try follows : answer by complement ; By using Stars and Bars:
$A+B+C+D=50$
Total number of ways to distribute $50$ prizes onto $4$ teams = $53C3$
1) Ways that $A \geq 16$ : $A'+B'+C'+D'=34 \to 37C3$
2) Ways that $B \geq 21$ : $A'+B'+C'+D'=29 \to 32C3$
{Intersection between 1&2 $A'+B'+C'+D'=13 \to 16C3$
Answer = $53C3 - (37C3 +32C3-16C3)$
Is my answer right?
If not, please help me understand my fault.
Thank you for your help
