They Must Be In Descending Order?

For any $5$ integers chosen from the set $\{1,2,3,4,....,19\}$ there is only one way of arranging them in descending order, thus the answer is $\dbinom{19}{5} = 11628$ .

If $N(a,b)$ is the number of ways of choosing $a$ positively increasing positive integers less than $b+1$ , as described in the problem, for $b=19$ . Then,

• $N(a,b) = 0$ if a > b
• $N(1,1) = 1$
• $N(1,b) = b$
• $N(a,b) = N(a-1,b-1) + N(a,b-1)$ otherwise

Using these formulae, $N(5,19) = \boxed{11628}$