a) Consider different places.
The first place can be filled in ways.
The second place can be filled in ways.
The third place can be filled in ways.
.........................................................................
The k-th place can be filled in ways.
Together the number of ways is ways.
ways.
This is called
b) When the places are indistinguishable, the number reduces by a factor of . Because places can be ordered in ways. That is the required ways is
ways.
c) Here . number of a's can be selected in ways. So the coefficient of is .
Thus,
The proof is complete.
1. Justification of the two formulas for permutation and combinations. (15 pts) a) Assume there are n different objects...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a...