How many numbers can be formed using digits 1 2 3 4 5 6 7 8 9 such that they are in increasing order(e.g. 12345, 578)? Help me in how to solve this problem please. I will be grateful to you :)
 A: The number of numbers with digits in increasing order, counting those with one digit all the way to nine digits, is simply $\binom{9}{1}+\binom{9}{2}+\ldots +\binom{9}{9}$.
A: Consider the string $123456789$. Observe that each of the $9$ digits in the string can safely be either included or not included, since failing to include a digit will not violate the increasing order of the digits. Hence, there are $2^9$ possible strings that contain ($0$ or $1$ or $2$ or ... or $9$) digits. However, we likely want to omit the case where there are $0$ digits, which leaves us with a final answer of:
$$
2^9 - 1 = 511
$$
A: Well, I'm assuming you can only use each digit once.
If you start with 9, you can only do the one digit.
If you start with 8, you can do 8, or 89.
If you start with 7, you can do 7, 78, 79, or 789.
If you start with 6, you can do 6, 67, 68, 69, 678, 679, 689, or 6789.
I'll do one more.
If you start with 5, you can do 5, 56, 57, 58, 59, 567, 568, 569, 578, 579, 589, 5678, 5679, 5689, 5789, 56789.
Do you see the pattern?
For each digit d, there are 2^(9-d) possibilities.  Giving 511, I think.
A: numbers with digits in increasing order where they are in a.p. With $r=1 ...$ If it's a $5$ digit no and no starts with $5$ , then the number would be $56789$ . .. As I explained the question properly , now m pretty sure , u urself would be able to solve d problem , one digit numbers would not be considered .. $2$ digit no.- $8$ $(12,23,34,45,56,67,78,89)$
$3$ digit no.- $7(123,234,345,456,567,678,789)$,
$4$ digit no.- $6$, $5$ digit no -$5$, $6$ digit no -$4$, $7$ digit no-$3$, $8$ digit no-$2$, $9$ digit no -$1(123456789)$ ,... So total $=1+2+3+•••+8=36 $
A: I think it's 256 as the number of possibilities keeps doubling for every starting number. Viz., the numbers starting with 9 is 1 possibility,the numbers starting with 8 is 2 possibilities, the numbers starting with 7 is 4 possibilities, starting with 6 is 8, the numbers starting with 5 is 16, the numbers starting with 4 is 32,the numbers starting with 3 is 64, the numbers starting with 2 is 128, so finally, the the numbers starting with 1 is 256. So I think 256 numbers can be formed.
