Answer:
15
Step-by-step explanation:
You want the number of rising numbers between 2020 and 2400.
A "rising number" has digits that strictly increase from left to right.
The smallest rising number greater than 2020 is 2345. The largest rising number less than 2400 is 2389.
In this number range, the first two digits must be 23, and the remaining two digits are distinct and chosen from {4, 5, 6, 7, 8, 9}. Once a pair of digits is chosen, it can be arranged in increasing order, so the number of rising numbers is the number of ways 2 can be chosen from 6.
C(6, 2) = 6!/(2!(6 -2)!) = 6·5/(2·1) = 15
There are 15 distinct integers between 2020 and 2400 that have their digits in increasing order.
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Additional comment
They are ...
2345, 2346, 2347, 2348, 2349,
2356, 2357, 2358, 2359, 2367,
2368, 2369, 2378, 2379, 2389