The problem is asking for the number of strictly increasing functions $f : \{ 1, 2, \dots, n \} \to \{ 1, 2, \dots, m \}$ where $n$ and $m$ are positive integers, and $f$ satisfies the condition $f(i) < f(j)$ whenever $i < j$.

In general, a strictly increasing function selects $n$ distinct elements from the set $\{ 1, 2, \dots, m \}$ and assigns them to $\{ 1, 2, \dots, n \}$ while preserving order. This means we are choosing $n$ distinct values from a total of $m$, and then ordering them in increasing order, which is automatically determined since the function must be strictly increasing.

The number of ways to choose $n$ distinct elements from $m$ elements is given by the binomial coefficient: $\binom{m}{n} = \frac{m!}{n!(m-n)!}$ Thus, the number of strictly increasing functions is $\binom{m}{n}$.

Since no values of $n$ and $m$ are provided in the question, we cannot specify the exact count. However, the phrasing "There are only three strictly increasing functions" is likely incorrect unless $m$ and $n$ are such that $\binom{m}{n} = 3$.

Would you like further clarification or a more specific example with values of $m$ and $n$?

Here are five related questions:

1. What is the difference between a strictly increasing function and a non-strictly increasing function?
2. How do you calculate binomial coefficients?
3. What are the conditions under which a function is called injective?
4. Can a strictly increasing function ever be constant?
5. How does the concept of "strictly increasing" change for real-valued functions?

Tip: Remember, a binomial coefficient counts combinations, and it represents how many ways you can choose $n$ items from $m$ without regard to order.