Скачать или смотреть CO16 Combinations, Binomial Coefficients, & Lattice Paths

An s-Combination is a subset of size s. The number of s-combinations of a set with t elements is a #binomial coefficient read "t choose s". We give the definition, a formula and its proof, several examples (including counting paths on a grid), a #recurrencerelation with its proof, and briefly explain why we call the triangle of binomial coefficients the Karaji-Jia triangle (as opposed to #pascaltriangle). Subscribe ‪@Shahriari‬ for more undergraduate Math videos.
00:00 Introduction
00:15 Definition: s-combinations, subsets of size s
01:26 Definition: "t choose s", binomial coefficients
01:55 Example of binomial coefficients
02:39 Special cases: "n choose 0", "n choose n", "n choose 1"
03:29 Derivation of a formula for binomial coefficients
07:24 The formula for binomial coefficients
07:36 Translating to Balls & Boxes
08:46 Example: Deal 13 cards to each of 4 players. What is the probability that each get exactly 3 picture cards?
12:16 Example: Number of ways of pairing 8 people into 4 couples
15:57 Example: Number of NE paths on a grid
19:39 Lemma: "n choose k" is equal to "n choose n-k"
20:11 Proof of Lemma
22:05 Recurrence relation for binomial coefficients
22:25 Proof of recurrence relation
23:38 Triangle of Binomial Coefficients
24:37 Pascal's Triangle or Karaji's triangle or Jia's triangle
25:19 Karaji's triangle
26:00 Jia's triangle

A series of lectures on introductory Combinatorics. This full course is based on my book
Shahriar Shahriari, An Invitation to Combinatorics, Cambridge University Press, 2022.
DOI: https://doi.org/10.1017/9781108568708

For an annotated list of available videos for Combinatorics see
https://pomona.box.com/s/by2ay2872avx...
YouTube Playlist:    • Combinatorics, An Invitation  

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA
Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.