I would approach this problem via inclusion-exclusion based on the events:
$M$: we have strictly fewer than $1$ M
selected
$I$: we have strictly fewer than $4$ I
's selected
$S$: we have strictly fewer than $4$ S
's selected
$P$: we have strictly fewer than $2$ P
's selected
The probability of failure then is $Pr(M\cup I\cup S\cup P) = Pr(M)+Pr(I)+Pr(S)+Pr(P)-Pr(M\cap I)-Pr(M\cap S)-Pr(M\cap P)-Pr(I\cap S)-\dots+\dots-Pr(M\cap I\cap S\cap P)$
You should be able to calculate each of $Pr(M),Pr(M\cap I),Pr(M\cap I\cap S),\dots$ and complete the calculations that way, though this will be rather tedious to do as you will have to potentially use case-work on the exact number of I
's and S
's appearing, etc...
For example, $Pr(M\cap I)$ is the probability no M
's and at most $3$ I
's were used. Breaking into cases based on the number of I
's used we have $Pr(M\cap I) = \binom{15}{0}\left(\frac{24}{26}\right)^{15} + \binom{15}{1}\left(\frac{1}{26}\right)\left(\frac{24}{26}\right)^{14}+\dots+\binom{15}{3}\left(\frac{1}{26}\right)^{3}\left(\frac{24}{26}\right)^{12}$
Doing it like this though, you might be breaking $M\cap I\cap S\cap P$ into $32$ events, which is incredibly tedious.