Answer: (1+x) n = sum nCr × x r.
- A (1+x)<sup>n</sup> = sum nCr × x<sup>r</sup>
- B x<sup>n</sup>, the generating function associated with a single term
- C n! x, a linear function mistaken for a generating series
- D e<sup>x</sup>, the exponential generating function for permutations, not combinations
Correct answer: A. (1+x)<sup>n</sup> = sum nCr × x<sup>r</sup>
Explanation: Binomial theorem: (1+x)ⁿ = Σᵣ₌₀ⁿ C(n,r)xʳ. The coefficient of xʳ is exactly C(n,r). This is the ordinary generating function for the sequence of binomial coefficients.
Counting tree for selecting 2 items from {A, B, C} without repetition: 3 x 2 = 6 ordered arrangements.
Concept context
Counting, arrangements, and selections