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Recall that for every non-negative integer n the factorial is defined as n! = 1·2... (n-1). n. In particular, 0! = 1 and n! = (n − 1)! - n for n ≥ 1. For every integer 0 ≤ k ≤n, we now define the binomial coefficient (2) as n k We moreover set = n! k!(n - k)!* = 0 if k>n> 0. Let 1 ≤ k ≤ n. Prove that (2) = (¹) + (2−¹).

Recall that for every non-negative integer n the factorial is defined as n! = 1·2... (n-1). n. In particular, 0! = 1 and n! = (n − 1)! - n for n ≥ 1. For every integer 0 ≤ k ≤n, we now define the binomial coefficient (2) as n k We moreover set = n! k!(n - k)!* = 0 if k>n> 0. Let 1 ≤ k ≤ n. Prove that (2) = (¹) + (2−¹).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Recall that for every non-negative integer n the factorial is defined as n! = 1·2... (n − 1). n. In particular, 0! = 1 and n! = (n − 1)! - n for n ≥ 1. For every integer 0 ≤ k ≤n, we now define the binomial coefficient (2) as (3) = We moreover set (2) = 0 if k > n ≥ 0. n! k! (n – k)!* Let 1 ≤ k ≤ n. Prove that (2) = (n=¹) + (R=1).
Transcribed Image Text:Recall that for every non-negative integer n the factorial is defined as n! = 1·2... (n − 1). n. In particular, 0! = 1 and n! = (n − 1)! - n for n ≥ 1. For every integer 0 ≤ k ≤n, we now define the binomial coefficient (2) as (3) = We moreover set (2) = 0 if k > n ≥ 0. n! k! (n – k)!* Let 1 ≤ k ≤ n. Prove that (2) = (n=¹) + (R=1).
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