Solution

The correct answer is 2600

Key PointsThis problem deals with integer partitions and combinations. A standard trick for these kinds of problems is to shift everything to the left side to make all the variables unconstrained.

• Firstly, let's subtract the constraints from each variable:
  • x' = x - 1
  • y' = y - 2
  • z' = z - 3
  • w' = w - 0
• We can then express the equation in terms of the transformed variables:
  • x' + y' + z' + w' = 29 - (1 + 2 + 3 + 0) => x' + y' + z' + w' = 23
• Instead of four positive integers x, y, z, and w satisfying the original equation, we are looking for four non-negative integers x', y', z', and w' satisfying the new equation.
• This amounts to partitioning 23 into four parts allowing that some of the parts could possibly be 0.
• The basic formula for combinations with repetitions (stars-and-bars method) is C(n + r - 1, r - 1), where "n" is the sum and "r" is the number of variables.
• So we have C(23 + 4 - 1, 4 - 1) = C(26, 3) = 26! / ((26-3)! * 3!) = 2600.

So the answer is: 2600