A positive integer is called a rising number if its digits form a strictly increasing sequence. For example, 1457 is a rising number, 3438 is not a rising number, and neither is 2334.
(a) How many three digit rising numbers have 3 as their middle digit?
(b) How many three digit rising numbers are there?
My efforts have yielded 12 for (a) – 1 and 2 for the first digit, and 4, 5, 6, 7, 8, 9 for the 3rd so $2 \cdot 6 = 12$ possibilities. Is this correct? What is the best method for (b)?
Best Answer
If you pick any $k$ distinct digits out of 9, there is exactly one way to make a rising number out of it
Hence, total number of rising numbers is $$\sum_{k=1}^9{9\choose k} = 2^9-1$$
EDIT
For 3 digit numbers, if you pick any 3 distinct digits, there is exactly one rising number corresponding to those digits - hence there is a one-to one mapping between number of ways of selecting three distinct digits, and the number of 3 digit rising numbers
Hence - answer is ${9 \choose 3}$