Question: Let n be a positive integer, and let x1+x2+ +xk = n be a given equation. A vector (x1, x2, . . . ,

Let n be a positive integer, and let x1+x2+· · ·+xk = n be a given equation.

A vector (x1, x2, . . . , xk) satisfying x1+x2+· · ·+xk = n is said to be a nonnegative integer solution of the equation if for each i, 1 ≤ i ≤ k, xi is a nonnegative integer. It is said to be a positive integer solution of the equation if for each i, 1 ≤ i ≤ k, xi is a positive integer.

(a) How many distinct nonnegative integer solutions does the equation x1+x2+· · ·+xk =

n have?

(b) How many distinct positive integer solutions does the equation x1 +x2 +· · ·+xk = n have?

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