Question

How many three-digits numbers can be generated from 1,2,3,4,5,6,7,8,9 such that the digits are in ascending order ?

Answer 1

For each combination of 9 things taken 3 at a time, there is just 1 way they can be arranged in ascending order. So the answer is: 9C3 or C(9,3) or or or = 84 ways, regardless of what notation your book and teacher uses for the combination of 9 things taken 3 at a time. Answer = 84 -------------------------------------------- If the problem had asked this instead: How many three-digit numbers can be generated from 1, 2, 3, 4, 5, 6, 7, 8, 9 such that the digits are in NON-DESCENDING order? it would have been a harder problem, for we would have to add on the number of ways to have numbers of the form XXX, XXY, and XYY, (such as 333, 559, and 677, respectively). There would be C(9,1)=9 numbers of the form XXX, C(9,2)=36 numbers of the form XXY, and C(9,2)=36 numbers of the form XYY, (because for each combination there is just 1 way they can be arranged in non-descending order.) Then the answer would have been 84+9+36+36=165. I threw in this second problem just in case that was what your teacher really meant, but I think it was just the first way.