I have a problem where there are three types of lottery tickets, i.e. standard, VIP, premium. Each ticket has different odds of winning certain prizes (Say jackpot probability is $40\%$ in premium, $20\%$ in VIP, $5\%$ in standard). A person gets $10$ standard tickets, $5$ VIP tickets, $10$ premium tickets. I need to calculate the probability of a person winning $5$ jackpots from all his $25$ tickets.
My solution:
Use binomial distribution probability formula to calculate the odds of getting $0,1,2,3,4$ jackpots from the $10$ standard tickets alone, then repeat for the $5$ VIP and $10$ premium tickets. This results in a kind of $5×3$ matrix. Then I calculate the probability of winning $0,1,2,3,4$ tickets overall in this manner:
\begin{align*}
P(0) &= P_{\text{standard}}(0)×P_{\text{VIP}}(0)×P_{\text{premium}}(0)\\
P(1) &= P_{\text{standard}}(1)×P_{\text{VIP}}(0)×P_{\text{premium}}(0) + P_{\text{standard}}(0)×P_{\text{VIP}}(1)×P_{\text{premium}}(0)\\
&\quad + P_{\text{standard}}(0)×P_{\text{VIP}}(0)×P_{\text{premium}}(1)\\
P(2) &= P_{\text{standard}}(2)×P_{\text{VIP}}(0)×P_{\text{premium}}(0) + P_{\text{standard}}(0)×P_{\text{VIP}}(2)×P_{\text{premium}}(0)\\
&\quad + P_{\text{standard}}(0)×P_{\text{VIP}}(0)×P_{\text{premium}}(2) + P_{\text{standard}}(1)×P_{\text{VIP}}(1)×P_{\text{premium}}(0)\\
&\quad + P_{\text{standard}}(0)×P_{\text{VIP}}(1)×P_{\text{premium}}(1) + P_{\text{standard}}(1)×P_{\text{VIP}}(0)×P_{\text{premium}}(1)
\end{align*}
and so on for $P(3)$ and $P(4)$. (Note: there will be ten terms in the expression for $P(3)$ and fifteen terms in the expression for $P(4)$.) Then I conclude that the probability of winning is
$$P(\geqslant 5)=1-P(0)-P(1)-P(2)-P(3)-P(4).$$
My question is whether my approach is correct or not, and whether there would be a faster way to solve the problem.