Find number of $20$-combinations od multiset $M=\{d^2,a^2,b^\infty,c^\infty\}$.
I tried solving this using inclusion-exclusion formula. First, I wrote $M$ as: $M=\{d^2,a^2,b^{20},c^{20}\}$, since I am looking for $20$-combinations of multiset $M$. The total number of $20$-combinations of multiset $M$, when ignoring repetitions is $\binom{4+20-1}{20}=1771$. Now let $A_1$ be $20$-combinations of multiset $M$ in which $d$ appears at least $3$ times. Same goes for $A_2$, which is the number of $20$-combinations of multiset $M$ in which $a$ appears at least $3$ times. In these two cases I am looking for $17$-combinations of multiset $M$, so $|A_1|=|A_2|=\binom{4+17-1}{17}=1140$.
Now, $A_3$ and $A_4$ would be $20$-combinations of multiset $M$ in which $b$ and $c$ appear at least $21$ times, so $|A_3|=|A_4|=0$.
$|A_1 \cap A_2|=\binom{4+14-1}{14}=680$, these combinations have at least $3$ elements of type $d$ and at least $3$ elements of type $a$.
$|A_1 \cap A_3|=|A_2 \cap A_3|=0$
$|A_1 \cap A_2 \cap A_3|=0$
Lastly, the number I am looking for is: $1771-((1140+1140+0+0)-(680+0+0)+0)=171$.
Is this correct?