Counting Theory: Permutations, Combinations, Inclusion-Exclusion, and Binomial, Study notes of Discrete Mathematics

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Counting

If a first task can be done in The sum rule5.1 The basics of counting

n

1

ways, second in

n

2

ways and

n if these tasks cannot be done at the same time then there are

1

n

2

ways to do either of them.

If

A

1

,... , A

n

are sets which are pairwise disjoint then

| A 1 ∪ A 2 ∪

∪ A n | = | A 1 | + | A 2 | +

• | A n |.

Basic Inclusion exclusion principle:

Two sets:

| A ∪ B | = | A | + | B

A

B

Three sets:

| A ∪ B ∪ C | = | A | + |

B

C

A

B

A

C

B

C

+ | A ∩ B ∩ C |

Four sets:

Theorem 15.2 The pigeonhole principle

If

N

objects are placed into

k

boxes, then there is

a box with at least

N k

objects.

Suppose every box contains at most Proof.

N k

1 objects.

Then the

total number of objects is at most

k

( ⌈

N k

< k

N

k

N.

This is the contradiction as the number of objects is

N

P

n, r

) - the number of

r

-permutations of

n

elements.

C

n, r

) - the number of

r

-combinations of

n

elements.

Theorem 2

For

< r

n

,

P

n, r

n

(

n

n

r

Theorem 3

For

0 ≤ r ≤ n ,

C

n, r

n

!

r

!(

n

r

)!

5.4 Binomial Coefficients

(

r n

)

C

n, r

Combinatorial proof

• an argument which proves an identity by

providing a bijection from a set

A

to

B

where

A

has the cardinality

equal to the LHS and

B

has the cardinality equal to the RHS.

(

n

r

)

n

∑

j

=

(

j

r

)

.

Theorem 4Binomial Theorem:

For a positive integer

n

and real numbers

x

and

y

,

we have

( x + y ) n = n

∑

j

(

j n

) x n − j y j =

(

0 n

)

x

n

(

1 n

) x n − 1 y +

(

n n

)

y

n

.