4.1.12 - The Binomial Theorem I

4.1 - Algebra - Expressions

Leaving Certificate Mathematics

Higher Level ONLY

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 1 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3!

= 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5!

= 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1!

= 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0!

= 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

Notation

n! = n × (n − 1)× (n − 2)× . . .× 3× 2× 1 . . . ”n factorial”

Examples:

3! = 3× 2× 1

= 6

5! = 5× 4× 3× 2× 1

= 120

1! = 1

0! = 1

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 2 / 4

We can choose r objects from n objects in the following way:

(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)

=5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12

= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1

= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

We can choose r objects from n objects in the following way:(nr

)= n!

r !(n−r)! . . . ”n choose r”

Example:

(5

2

)=

5!

2!(5− 2)!

=5!

2!× 3!

=5× 4× 3× 2× 1

2× 1× 3× 2× 1

=120

12= 10

(5

2

)=

5× 4× 3× 2× 1

2× 1× 3× 2× 1

=5× 4

2× 1= 10

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 3 / 4

In general,(nr

)= n×(n−1)×...×(n−r+2)×(n−r+1)

r×(r−1)×...×2×1

Example: (10

3

)=

10× 9× 8

3× 2× 1

= 120

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 4 / 4

In general,(nr

)= n×(n−1)×...×(n−r+2)×(n−r+1)

r×(r−1)×...×2×1

Example:

(10

3

)=

10× 9× 8

3× 2× 1

= 120

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 4 / 4

In general,(nr

)= n×(n−1)×...×(n−r+2)×(n−r+1)

r×(r−1)×...×2×1

Example: (10

3

)

=10× 9× 8

3× 2× 1

= 120

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 4 / 4

In general,(nr

)= n×(n−1)×...×(n−r+2)×(n−r+1)

r×(r−1)×...×2×1

Example: (10

3

)=

10× 9× 8

3× 2× 1

= 120

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 4 / 4

In general,(nr

)= n×(n−1)×...×(n−r+2)×(n−r+1)

r×(r−1)×...×2×1

Example: (10

3

)=

10× 9× 8

3× 2× 1

= 120

4.1 - Algebra - Expressions 4.1.12 - The Binomial Theorem I Higher Level ONLY 4 / 4