How many integral solutions are there to x+y+z+w=29 ,where x≥1, y≥2, z≥3x and…

First, address the likely typo in the constraint z ≥ 3x by assuming it means z ≥ 3 to match standard problem types.

Transform variables to remove lower bounds: let x' = x-1, y' = y-2, z' = z-3, and w' = w. The equation becomes x'+y'+z'+w' = 23.

Apply the stars and bars formula C(n+k-1, k-1) with n=23 and k=4 variables. This results in C(26, 3).

Calculate the combination: (26 × 25 × 24) / 6 = 2600 integral solutions.