Inclusion-Exclusion Principle with Summations and combinations

Question

I am trying to understand this solution. I understand how they derived P(Aj), P(Aj1,Aj2), etc. After using the inclusion exclusion principle I get lost. How did they go from the inclusion-exclusion expression using summations to the expression with combinations? I'm struggling to understand how they are the same.

Answer 1

In the first sum, the summand is $\left(\frac{N-1}{N}\right)^n$, regardless of the value of $j$. Hence the sum reduces to $N\left(\frac{N-1}{N}\right)^n$ as there are $N$ different values of $j$.

In the second sum, the summand is $\left(\frac{N-2}{N}\right)^n$, regardless of the values of $j_1$ and $j_2$. This sum can therefore reduced to the number of terms multiplied by the common summand. The number of terms is the number of ways of choosing the two values of $j_1$ and $j_2$. The number ways of doing this is the number of ways of choosing two numbers from the set $\{1,2,\ldots,N\}$, which is $\binom{N}{2}$. The smaller of the chosen numbers will be $j_1$ and the larger will be $j_2$.

All of the sums are handled in this way.