Find the number of solutions of the equation x1 x2 x3 x4

Find the number of solutions of the equation x1 + x2 + x3 + x4 = 21, where xi , i = 1, 2, 3, 4, are nonnegative integers such that 1 < x1 3, 2 x2 < 5, x3 5, and x4 < 9

Solution

x1 => (x^k) [k = 2 to 3]

x2 => (x^k) [k = 2 to 4]

x3 => (x^k) [k = 0 to 5]

x4=> (x^k) [k = 0 to 8]

(x^k)[k = 2 to 3] * (x^k)[k = 2 to 4] * (x^k)[k = 0 to 5] * {(x^k)[k = 0 to 8]}

so on expanding this equation we can see that there is no term contaning x²¹

hence there are no non-negative integer solutions.