Sum of roll of N dice

[Joe Horn]

THE QUESTION: Is there a direct way to return the sum of a roll of N dice without generating N random numbers and adding them together?

BACKGROUND: In a recent Facebook discussion, it was suggested that you simply generate a random number uniformly between N (a roll of all 1's) and 6N (a roll of all 6's). But that assumes that every total is equally probable, which is not true.

For example, the sum of a roll of 4 dice can be anything between 4 and 24... but not every sum has equal probability of occurring. Since there are 4 dice, each of which can have 6 values, there are 6^4 (1296) possible ways to roll 4 dice. There is only 1 way to roll a total of 4:

{ 1 1 1 1 }

so rolling a total of 4 has a probability of 1/1296. But there are 4 ways to roll a total of 5:

{ 1 1 1 2 }
{ 1 1 2 1 }
{ 1 2 1 1 }
{ 2 1 1 1 }

so rolling a total of 5 has a probability of 4/1296. There are 10 ways to roll a total of 6:

{ 1 1 1 3 }
{ 1 1 3 1 }
{ 1 3 1 1 }
{ 3 1 1 1 }
{ 1 1 2 2 }
{ 1 2 1 2 }
{ 2 1 1 2 }
{ 1 2 2 1 }
{ 2 1 2 1 }
{ 2 2 1 1 }

so rolling a total of 6 has a probability of 10/1296. Different totals have different probabilities. How can this be taken into account when simulating throwing N dice directly, without summing N random integers between 1 and 6?