One can apply Binomial distribution if, p probability of success is constant for all events that are independent. However, if p is the same but the events differ then is it still applicable?
Example; Typical elementary questions on this ask ;
What is the probability that exactly 3 four's will show up if a die is rolled 6 times
Q1) Given each face has equal probability. Is Binominal distribution still applicable if the event changes but p is the same?
Example: 1 four , 1 six , 1 five if the die is rolled 6 times?
Q2) if p is different for some events. So p1 p2. Is there any way to split it and combine?
Example: An exam has 3 - single choice question (SCQ. p1=0.25) and 4 - MCQs (p2 = 0.1). What is the probability we get 2 SCQ right and 1 MCQ right?
Q1
This is known as a multinomial distribution From your clarification, it is clear that you can divide the outcomes into getting $4,5,6$, and any other three with probabilities of $\frac16,\frac16,\frac16,\; and\; \frac36$ respectively
Instead of the binomial coefficient, you will have the multinomial coefficient $\large\binom{6}{1,1,1,3} = \large\frac{6!}{1!1!1!3!}$
The formula will then be $\frac{6!}{1!1!1!3!}\cdot(\frac1 6)^3(\frac3 6)^3$
Q2
Yes you can use the binomial distribution for the two parts. Since now there are two separate Bernoulli trials, and you want the Pr of 2 SCQ right and 1 MCQ right, and the two trials are independent,
$Pr = [\binom32*0.25^2*0.75^1]*[\binom41(0.1^1)(0.9^3)]$