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Infinitely many. The normal distribution is applicable to a continuous variable whose domain is the whole of the real numbers.
The binomial distribution is a discrete probability distribution. The number of possible outcomes depends on the number of possible successes in a given trial. For the Poisson distribution there are Infinitely many.
If the numbers (or symbols) are all different then 10 outcomes.
Possible outcomes of a single dice are 6 ( 1,2,3,4,5,6) So if 5 such dices are rolled then the number of possible outcomes are 6 mulitiplied by 6 five times. 6x6x6x6x6x6=46656 possible outcomes.
8 outcomes are possible in this situtation. You just have to multiply 4 by 2 to get the answer.
No. The binomial distribution (discrete) or uniform distribution (discrete or continuous) are symmetrical but they are not normal. There are others.