Combinatorics

Question 1

What does 10! means

Answer 1

It’s the number of arrangements of 10 elements IN DIFFERENT ORDERS
(order matters)

Question 2

What does 10C5 means

Answer 2

It’s the number of groups of 5 that can be created from 10 elements, if order doesn’t matter
(109876) / 5!

Question 3

What is 10* 9* 8* 7* 6

Answer 3

It’s the number of groups of 5 elements that can be created from 10 elements IN DIFFERENT ORDERS
(order matters)

Question 4

What is a * b* c in combinatorics?

Answer 4

The number of groups that can be created from sets A, B and C, if ORDER MATTERS

Question 5

What is a * b* c / 3! in combinatorics?

Answer 5

The number of groups that can be created from sets A, B and C, if ORDER DOESN’T Matter (eliminated repetition)

Question 6

What to do on a “at least one” combinatorics question?

Answer 6

1) draw the scheme
2) understand how you can create the group
3) calcualte the complementary groups (the ones you don’t want)
4) calculate total - complement

APPROACH FROM COMPLEMENTARY GROUPS

Question 7

What are the options of ordering in combinatorics?

Answer 7

Order matters: A,B <> B, A
Order doesn’t amtter: A,B = B,A

Question 7

How to remove repetitions in combinatorics?

Answer 7

If order doesn’t matter, you remove repetition by dividing by the number of repeated groups

In a group of 3, there are 3! possible orders

5C3, you could have 543 groups, but then 3! would be same elements in different orders

Question 7

How to deal with constraints?

Answer 7

Tie the elements together
If on a group of 5 A must not be in corners and be near B, then

3 * 2 * 3 * 2 * 1

If on a group of 5 A must be near B, then

3 * 2 * 3 * 2 * 1
+
2 * 1 * 3 * 2 * 1

Question 7

If you have 5 books A and 5 books B, how many groups of 3 can you create with at least one A and one B?

Answer 7

10C3 - 5C3 - 5C3

total - groups with only A - groups with only B = groups with at least one of each