An experiment has $n$ outcomes, which is repeated $r$ times. Let $\{x_1,x_2, \ldots, x_n\}$ record the occurrences of $\{A_1,A_2, \ldots, A_n\}$. Show that the number of sample points is $$\binom{n+r-1}{r-1}.$$
Approach 1: In this question, the experiment is carried out $r$ times. I represent the set of outcomes by set $S$.
If the experiment is repeated $2$ times, the number of sample points would be
$$\{A_1,A_2, \ldots, A_n\} \times \{A_1,A_2, \ldots, A_n\}$$
Clearly, each $(A_i, A_j)$ is an ordered pair, and the number of sample points is $n^2$.
If the experiment is performed $r$ times, we get $\vert \Omega \vert= n^r$.
Approach 2:
Here it is given that $\{x_1,x_2, \ldots, x_n\}$ record the outcomes, so we could think that each of $\{A_1,A_2, \ldots, A_n\}$ would have happened $0$ or more times.
So we could write $$\{x_1+x_2+ \ldots + x_n =r\}$$ so the number of sample points would be same as the number of non negative integral solutions of the above equation which is $$\binom{r+n-1}{n-1}$$ which can be rewritten as $$\binom{n+r-1}{r}$$
In either approach, I am unable to find the correct answer, if I consider approach 2, I am trying to trace back the sample points with respect to the number of outcomes of each sample point in the original experiment. However, this leads to missing the count of other $n$ ordered tuples sample point which are counted in approach 1. Where am I going wrong?
Your second solution is correct.
The way I read this $A_1, A_2, \ldots, A_n$ are the possible outcomes of the experiment. If we let $x_i$ denote the number of times the outcome $A_i$, $1 \leq i \leq n$, occurs in $r$ trials of the experiment, then $$x_1 + x_2 + \ldots + x_n = r$$ is an equation in the nonnegative integers since outcome $A_i$ is not guaranteed to occur. Since a particular solution of the equation corresponds to the placement of $n - 1$ addition signs in a row of $r$ ones, the number of solutions of the equation in the nonnegative integers is $$\binom{r + n - 1}{n - 1} = \binom{r + n - 1}{r}$$ since we must choose which $n - 1$ of the $r + n - 1$ positions required for $r$ ones and $n - 1$ addition signs will be filled with addition signs or, alternatively, which $r$ of the $r + n - 1$ positions will be filled with ones.