Let set S={1,2,3,4,5}
// we can pick any element any number of times.
a) repetition is not allowed and order of picking matters.
repetition not allowed means we can't pick any element more than one time in a single extraction of elements.
like we have to pick 2 elements then those can be (1,2)(2,1)(1,3)(3,1)(1,4)(4,1)(1,5)(5,1)(2,3)(3,2)(2,4)(4,2)(2,5)(5,2)(3,4)(4,3)(3,5)(5,3)(4,5)(5,4) so 20 ways (nCr)*r!
b)repetition is allowed and order of picking doesn't matter.
Here repetition is allowed so (1,1)(2,2)(3,3)(4,4)(5,5) can we out of those 2 we're picking.here order doesn't matter means (1,2) and (2,1) is same. so total=10+5=15 ways.
c)repetition is allowed and order of picking matters.
so in this case simply there would be n choices for each place out of r => n^{r}
d) repetition is not allowed but order of pickings doesn't matter.
(1,2)(1,3)(1,4)(1,5)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)
simply nCr ways