Question 48
a. Given that $(1+x)^{p-2} = 1 + a_1x + a_2x^2 + \dots + a_{p-2}x^{p-2}$ where $p$ is an odd prime number, prove that $a_1 + 2, a_2 - 3, a_3 + 4, \dots, a_{p-3} - (p-2)$ and $a_{p-2} + (p-1)$ are all multiples of $p$.
b. How many integer solutions are there to the equation $x_1 + x_2 + x_3 + x_4 = 100$ where $-1 \le x_1 \le 10, 0 \le x_2 \le 11, 1 \le x_3 \le 12$ and $x_4 \ge 50$?
b. How many integer solutions are there to the equation $x_1 + x_2 + x_3 + x_4 = 100$ where $-1 \le x_1 \le 10, 0 \le x_2 \le 11, 1 \le x_3 \le 12$ and $x_4 \ge 50$?