Help with understanding Bose-Einstein value

#1
I can't seem to understand the explanation to this question: "How many ways are there to choose k times from a set of n objects with replacement?"

The explanation is in the image below. I don't understand why it is (n + k -1) choose k. How does he get to (n + k - 1)? I understand what he's saying about n - 1 |'s and k *'s. I would appreciate any advice on understanding this. Thank you!
IMG_5651.jpg
 
#3
#4
its sort of weird to me they named it the 'bose-einstein value'. The argument was stolen from combinatorics time in memorial, probably bernoulli or newton i think is associated with negative binomials?