Digits in increasing order

Text

How many strictly increasing $4$-digit strings, i.e. of the form $abcd$ with $a<b<c<d$, can be formed using the digits from $0$ to $9$?

Solution

Once a set of $4$ distinct digits is chosen, there is exactly one way to arrange them in increasing order. So counting increasing strings is the same as counting the subsets of $4$ distinct digits taken from $10$: these are combinations, $N = \binom{10}{4} = \frac{10\cdot 9\cdot 8\cdot 7}{4!} = \frac{5040}{24} = \boxed{210}.$