How many five digit numbers of different digits can be made in which digits are in ascending order?
Well, when i was trying to solve the problem, i tried taking cases fixing 1 in the first place and so on.. However as it is evident, the 2nd place then maybe filled by 2,3,4,5,6 which implies 5 ways. If we count 5 ways for this place, we cannot say with certainty if 3,4,5,6,7 may enter the 3rd place or not, as if the 2nd place is occupied by 6, no other possibility remains.
Then, i also saw a pattern being followed in the possible numbers which may or may not be filled in a given place, excluding one possibility after each step.. However, was unable to calculate the same.
[Math] Five digit number with digits in ascending order
combinationscombinatoricspermutations
Best Answer
The answer is $\displaystyle{9\choose 5} = 126$:
Each number $abcde$ is the same as choosing a subset $\{a,b,c,d,e\}$ of $\{1,2,...,9\}$.
For example, the number $24579$ is the same as set $\{2,4,5,7,9\}$ and vice versa, a set $\{1,3,4,5,8\}$ determines a number $13458$.