How many five digit number can be formed such that their numbers counted from left to right creates a decreasing sequence?
Numbers $= {0,1,2,3,4,5,6,7,8,9}$
Example: $54321$ and $96310$
If we would count the amount of five digit numbers that can be formed such that their number counted from left to right is INCREASING we would get:
$\binom{9}{5}=126$
Since the amount of distinct set can be sorted so $n_1>n_2>n_3>n_4>n_5$
Were $n_1$ is the digit with the highest value.
My attempt at the real question in hand is the following:
We sort every digit so that $n_1<n_2<n_3<n_4<n_5$
We would now get $\binom{10}{5}=252$ decreasing five digit number.
This is clearly wrong since I don't account for the zero.
I need to divide this problem into different parts in some way, any got any suggestions?
Thanks beforehand!
Best Answer
Hint the answer will be ${10\choose 5} $ because when you select any $5$ digits there will always be a sequence st all the digits form a decreasing order.