I am trying to derive an analytical formula for this problem.

I have 10 numbers (n=10) and I choose k from them (ranging from 1 to 10 at the end). Let k = 4 for this example's sake.

I want to find the probability that,

all the numbers are that I choose are different,

$\displaystyle 10*{1}/{10}*{9}/{10}*{8}/{10}*{7}/{10}$

all the numbers that I choose are thesame,

$\displaystyle 10*{1}/{10}*{1}/{10}*{1}/{10}*{1}/{10}$

and the other respective combinations ?

I think that the probability of choosing two distinct numbers, is

1) the probability of choosing a sequence like xx yy

2) the probability of choosing a sequence like xxx y

such that

$\displaystyle 10*{1}/{10}*{1}/{10}*{9}/{10}*{1}/{10}$

and

$\displaystyle 10*{1}/{10}*{1}/{10}*{1}/{10}*{9}/{10}$

and the probability of choosing three distinct numbers should be given by

$\displaystyle 10*{1}/{10}*{9}/{10}*{8}/{10}*{2}/{10}$

i know that these calculations have to be scaled by multiplying with the possible number of occurrences (combinations), but I'm a bit lost on how to continue!

I would also gladly accept and alternative methods to tackle this solution

thanks