Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Combinatorics and Graph Theory · Problem 1.8.5
Fundamentals — David Guichard
Mathematics — Combinatorics · stirling-numbers-falling-factorial
Problem Show that x n ‾ = ∏ k = 0 n − 1 ( x − k ) = ∑ i = 0 n s ( n , i ) x i x^{\underline{n}} = \prod_{k=0}^{n-1}(x-k) = \sum_{i=0}^{n} s(n,i)x^i x n = ∏ k = 0 n − 1 ( x − k ) = ∑ i = 0 n s ( n , i ) x i n ≥ 1 n \geq 1 n ≥ 1 x n ‾ x^{\underline{n}} x n x n ‾ = ∏ k = 0 n − 1 ( x + k ) x^{\overline{n}} = \prod_{k=0}^{n-1}(x+k) x n = ∏ k = 0 n − 1 ( x + k ) x n ‾ x^{\overline{n}} x n Browse the problem library