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Use the separator method.

Let the 5-unit sum = DDDDD.

We can use these five D's to represent any integer in which the sum of the digits is 5:
DDDDD = 5.
DDDD D = 41.
D DDD D = 131.

To be less than 10,000, the integer can have at most 4 digits.
Thus, the five D's can be separated at most into 4 groups. For example:
D D D DD = 1112.

Thus, there are at most 3 separators dividing the five D's.
Let the 3 separators = |||.

Thus:
|||DDDDD = 5.
||DDDD|D = 41.
|D|DDD|D = 131.
D|D|D|DD = 1112.

Thus, to count the number of integers, we need to count the number of ways to arrange DDDDD|||.
Number of ways to arrange DDDDD||| = 8!/(5!3!) = 56.

The correct answer is C.

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The separator method can used to count the number of ways to distribute identical elements.

Four identical balls are thrown into the air. Three children are playing a game in which each child holds a barrel that is to be used to catch the balls. If at the end of the game all of the balls have been caught, in how many ways could the four balls be distributed among the three children?

Using the separator method, we'd have four balls (BBBB) and two separators (||), since the balls can be split into a maximum of 3 divisions.

Number of ways to arrange BBBB|| = 6!/(4!2!) = 15.

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Mitch Hunt
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If you revisit my post above, you'll note that I've edited the question to make its intention clearer.

The separator method counts the number of ways in which the four balls can be distributed among the 3 children:
Number of ways to arrange BBBB|| = 6!/4!2! = 15.

Here are all the possible distributions:
1 child catches all 4 balls, the other 2 children each catch 0 balls:
Number of ways to arrange 4-0-0 = 3.

1 child catches 3 balls, 1 child catches 1 ball, 1 child catches 0 balls:
Number of ways to arrange 3-1-0 = 6.

2 children each catch 2 balls, 1 child catches 0 balls:
Number of ways to arrange 2-2-0 = 3.

1 child catches 2 balls, the other 2 children each catch 1 ball:
Number of ways to arrange 2-1-1 = 3.

Total number of ways to distribute the balls = 3+6+3+3 = 15.

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Mitch Hunt
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The separator method is used to count the number of ways in which identical elements can be distributed.
Since in the problem above the sum of the digits must be 5, the separator method can be used to count the number of ways in which the 5-unit sum (DDDDD) can be distributed among the 1,2,3 or 4 digits.

To illustrate:
|D|DD|DD = a distribution of 0-1-2-2, representing the number 122.
|DD|D|DD = a distribution of 0-2-1-2, representing the number 212.
|DD|DD|D = a distribution of 0-2-2-1, representing the number 221.

Notice that the distributions above account for all the different ways in which the digits 1,2,2 can be ordered in a 3-digit integer.

Thus, by using the separator method to count all the different ways in which the 5-unit sum can be distributed, we account for every possible sequence of digits.

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Mitch Hunt
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Number of separators = Maximum number of distributions - 1.

In the digits problem above, the sum can be distributed among up to 4 digits, so the maximum number of distributions is 4.
Number of separators = 4-1 = 3.

In the game problem that I posted, the balls can be distributed among up to 3 children, so the maximum number of distributions is 3.
Number of separators = 3-1 = 2.

_________________
Mitch Hunt
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GMATGuruNY@gmail.com
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