what is the relation between probability vs permutations and combinations?

Question

Can some one explain what is the relation between probability vs permutations and combinations? How can we relate these two concepts? For toss and die i am able to understand but for set of numbers a little bit of confusion occurring.

Answer 1

Probability is -fundamentally- about sizes of certain sets. (When we say that some event has probability a half, we actually mean that the set of outcomes that constitute that event have a "size" of 1/2.) Permutations and combinations allow you to count, i.e. determine the sizes of certain sets.

EDIT: After the comment by the OP, I decided to add an example: given the question, I expect OP wants an example other than die and coin tosses, so here is one:

Let's say we are tossing N balls to three buckets: A, B, and C, and each ball has an equal chance of landing in each bucket.

Now, about this experiment, we can ask: "What is the probability that there are k balls in bucket A?"

To do this question, we should think about how many outcomes we could have had: We can label each outcome as "First ball went to A, second ball went to A, third ball went to C, ..., Nth ball went to A". (For compactness, we can represent this sentence as (A,A,C,...,A).) Now, how many such outcomes are there? For each ball, there are three choices, and there are N balls, so there are $3^N$ outcomes. We are going to "normalize" all the sets by this factor, so that the set that contains all the outcomes has size 1.

Now, how many of these outcomes do we care about? We only care about those that have exactly $k$ $A$s in them. How many ways can we choose exactly $k$ balls out of $N$? That is $\binom{N}{k}$. Now, for all the other balls, they either went to bucket B or C, so they each had 2 choices, so that gives us another $2^{N-k}$. So, our net size of the set that we care about is:

$\frac{\binom{N}{k}2^{N-k}}{3^N}$.

(You will see this as a Binomial distribution in future, but it follows directly from combinatorics as you can see above.)