Problem 18 - Entrance Test

This is a stars and bars problem with a lower bound for each variable. We have the equation x + y + z + w = 15, with the conditions x >= 2, y >= 2, z >= 2, w >= 2. To use the standard stars and bars formula (which applies to non-negative integers), we make a substitution. Let x' = x - 2, y' = y - 2, z' = z - 2, w' = w - 2. Now, x', y', z', w' must be non-negative integers (x', y', z', w' >= 0). Substitute these into the original equation: (x' + 2) + (y' + 2) + (z' + 2) + (w' + 2) = 15 x' + y' + z' + w' + 8 = 15 x' + y' + z' + w' = 15 - 8 x' + y' + z' + w' = 7. Now we need to find the number of non-negative integer solutions to x' + y' + z' + w' = 7. Using the stars and bars formula, for an equation of the form a1 + a2 + ... + ak = n (where ai >= 0), the number of solutions is C(n + k - 1, k - 1) or C(n + k - 1, n). Here, n = 7 (the sum) and k = 4 (the number of variables). Number of solutions = C(7 + 4 - 1, 4 - 1) = C(10, 3) C(10, 3) = 10! / (3! * (10 - 3)!) = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1) = (10 * 3 * 4) (after canceling 9/3 and 8/2) = 120. There are 120 distinct positive integer solutions.