Can apply the stars and bars formula calculate the number solutions

To find the number of integer solutions to the equation X1 + X2 + X3 + X4 = 38, subject to the conditions X1 ≥ 5, X2 ≥ 6, X3 ≥ 3, and X4 ≥ 5, we can use the concept of stars and bars.

First, let's introduce new variables Y1, Y2, Y3, and Y4, where Y1 = X1 - 5, Y2 = X2 - 6, Y3 = X3 - 3, and Y4 = X4 - 5. This allows us to rewrite the equation as Y1 + Y2 + Y3 + Y4 = 19, where Y1, Y2, Y3, and Y4 are non-negative integers.