Combinatorics

What is combinatorics (in simple terms)?
Combinatorics is the science of how to combine different objects.

Binomial theorem

Recommended books:

Set in combinatorics - A = {a1, a2, a3, ..., n}

Number of elements in set (power) 0:07:42?
Number of elements in set - A = |A| = n

A = {a1, a2, a3, ..., n}
B = {b1, b2, b3, ..., m}, no duplicates, number of items can be other

Number of ways to select 1 object from A or 1 object from B ?
|A| + |B| = n + m

A = {a1, a2, a3, ..., n}
B = {b1, b2, b3, ..., m}, can contain duplicates

Number of ways to select 1 object from A and then 1 object from B (2 selection)?
|A| * |B| = n * m

a1b1, a1b2, a1b3, a1bm
a2b1, a2b2, a2b3, a2bm
anb1, anb2, anb3, anbm
We can also multiply many sets together, (|A| * |B|) * |C|

If n items are put into m containers, with n > m, then at least one container must contain more than one item.

A = {a1, a2, a3, ..., n}
k (natural numbers + 0) <= n

How to select k items from A (possible ways)?
So here the list: We can select k items sequentially, with or without removing items from set:

k-placements without repetition (cut, where k is number of items)
k-placements with repetition (can be without repetition) We can select k items simultaneously, with or without removing items from set:
k-combinations without repetition
k-combinations with repetition (can be without repetition)