39660

I did the math practically using buttons and putting them in 10 bowls.I used 44 buttons and tried to put them into the bowls so that each bowl had a different number of buttons.That didn't work so I used 45 buttons instead and tried to put them in the ten bowls. I decided that one of the bowls could have no buttons in it. I found that this time I was successful.

So then I did the math on paper and saw that 0-9 added up to 45. 45 is the minimum number of coins she can have in 10 pockets because 0-9 is the lowest set of ten different positive numbers that we can have.