How do I do combinatoric algebra without tedious factorial multiplication and division?

Question

I find combinatoric algebra very non-intuitive. I'm talking about Pascal's Identity $n\geq r$, $$ \binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}. $$

I understand the tedious proof of the theorem but what's a trick for understanding combinatoric algebra in general? I can't eyeball and decompose a binomial without memorizing the formulas or doing tedious factorial multiplication and division.

It's never obvious how combinatoric algebra works:

EXample:

The trick is to attend to what these numbers (here binomial coefficients and Stirling numbers) count. — Angina Seng, Jul 15 '17 at 18:18
You should probably describe what is the "tedious combinatoric proof" of that property of binomial numbers. I simply cannot imagine what "tedious factorial multiplication and division" you are doing... — Mariano Suárez-Álvarez, Jul 15 '17 at 18:20
Hi OP! In regards to your specific question about Pascal's identity, Wikipedia gives a "combinatorial proof". Searching for more examples of "combinatorial proof" should give you more examples of the kind of high-level arguments you're looking for. (On this site, there is an user Brian M. Scott who does quite a lot of them - for example, here.) — Chris, Jul 15 '17 at 18:51
And on a related note, you might also appreciate the answers to this question. (Including mine!) — Chris, Jul 15 '17 at 18:53

score 1 · Answer 1 · answered Jul 15 '17 at 18:15

1

For the second one:
You want to partition $\{1,2,\cdots ,n,n+1\}$ into $m+1$ non empty blocks(RHS). What if you first take out the elements that are in the block of $n+1$(Say there are $k$ elements) and then you partition the rest?(LHS)

I agree, "combinatorial algebra" is hard but that's why combinatorics uses two powerful concepts: Intuition on the combinatorial description (with interesting principles as sum, multiplication and bijection) and generating functions.

answered Jul 15 '17 at 18:15

Phicar

14,827

combinatorics have to do with cardinality and functions? – Jul 15 '17 at 18:27
Yes, indeed. Define the numbers $n$ as cardinals of the set $[n]={1,2,\cdots ,n}.$ For example, $\binom{n}{k}=|\binom{[n]}{k}|,$ where $$\binom{[n]}{k}={A\subseteq [n]:|A|=k}.$$ Which encodes the notion of choose $k$ from $n$. – Phicar Jul 15 '17 at 19:39

score 0 · Answer 2 · answered Jul 15 '17 at 18:10

0

Guide:

Your goal is to pick $r$ items form $n+1$ items.

suppose you pick the first item, how many more items do you still have to pick?

what if you did not pick the first item?

answered Jul 15 '17 at 18:10

Siong Thye Goh

153,832

How do I do combinatoric algebra without tedious factorial multiplication and division?

2 Answers2