SOLUTION: How many 3-digit numbers are there for which the sum of the digits of the number is 24? (A)3 (B)6 (C)7 (D)10 (E)18

The smallest one will have the 2nd and 3rd digits as large as
possible, which is both 9's, and therefore the hundredths
digiuts has to be 6, so 699 is the smallest.

 1.  699

Then the next larger one can be made by
increasing the 1st digit by 1 and decreasing the
third digit by 1.

 2.  789

The next larger one can be made by increasing the
the 2nd digit by 1 and decreasing the 3rd digit
by one.

 3.  798

The next larger one can be made increasing the 1st
digit by 1, and making the 3rd digit large as possible,
and that leaves 7 for the 2nd digit:

 4.  879

The next larger one can be made by increasing the 2nd
digit by 1 and decreasing the 3rd by 1:

 5.  888

The next larger one can be made by increasing the 2nd
by 1 and decreasing the 3rd digit by 1:

 6.  897

The next larger one will have to have to have a 9 as the
1st digit. We make the 3rd digit as large as possible, so
it will be 9. That leaves 6 for the 2nd digit:

 7.  969

The next larger one is gotten by increasing the 2nd digit
by 1 and decreasing the 3rd digit by 1. 

 8.  978

The next larger one is gotten by increasing the 2nd digit
by 1 and decreasing the 3rd digit by 1.

 9.  987

The only thing left to do is increase the 2nd digit by 1
and decrease the 3rd digit by 1.

10.  996

That's the largest one possible.

So the answer is "There are 10."

Edwin