A lottery has $N$ tickets numbered $1, 2, \dots, N$ with $N$ unknown. A random sampling of 5 winning tickets is numbered, say, $\{a, b, c, d, e\}\in[1, N]$.
The question I am trying to solve gives $a$ through $e$ explicitly, but I would simply like a hint on how we can go about estimating $N$.
On
HINT
The likelihood function is the pmf for the data viewed as a function of $N.$ Assuming the tickets are sampled with replacement, the pmf is is $\frac{1}{N^5}$ provided that all five of the ticket numbers are in $1,\ldots ,N$ and is zero otherwise. So you need to find the value of $N$ that maximizes this for the given sample.
EDIT
Michael Hardy gives a clarification in the comments in case my hint didn't adequately point out this important subtlety.
Here's a solution I was given.
For numbered tickets $\{a, b, c, d, e\}\in\mathbb{N}$ with $0<a<b<c<d<e$, the likelihood function is proportional to the probability of the observed data. There are ${N \choose 5}$ possible samples of size $5$ chosen without replacement from $N$ tickets. Each sample is equally likely. Therefore $$ P(\text{data}; N) = \frac{1}{{N \choose 5}}; N\ge e $$ This will be a maximum when $N \choose 5$ is as small as possible. But $N\ge e$, so the smallest possible value for $N$ is $e$, and so $\hat N=e$ is the maximum likelihood estimate of $N$.