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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
