a. Expand (1 + x)". b. Differentiate both sides of the equation from part (a) with respect to x to obtain n(1 + x)"-1 = C(n, 1) + 2C(n, 2) x + 3C(n, 3)x2 + .. . + nc(n, n)x-1 c. Prove that C(n, 1) + 2C(n, 2) + 3C(n, 3) + . . . + nC(n, n) = n2"-1 d. Prove that C(n, 1) - 2C(n, 2) + 3C(n, 3) - 4C(n, 4) + . . . + (-1)"-nc(n, n) = 0 22. (Requires calculus) a. Prove that C(n, 0) + C(n, 1) + C(n, 2) + ... + C(n, n) n + 1 n + b. Prove that C(n, 1) + C(n, 2) + .. . + (-1 )" C(n, n) n + 1 C(n, 0) - n + 1 (Hint: Integrate both sides of the equation from part (a) of Exercise 21.)