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Factorials for Binomial Expansion

Factorial Quotients for Binomial Expansion

(

)

n

0

=

(

)

n

n

=

n!

0! n!

=

1

=

1

1

(

)

n

1

=

(

)

n

n – 1

=

=

n (n – 1)!

1! (n – 1)!

=

n!

1! (n – 1)!

n (n – 1)!

(n – 1)!

=

n

Need to review factorial basics?
If so, then click: Understanding Factorials

(

)

2

1

=

2

(

)

3

1

=

(

)

3

2

=

3

(

)

4

1

=

(

)

4

3

=

4

and so on.

(

)

4

2

=

4!

2! 2!

=

4 ∙ 3

2 ∙ 1

=

6

(

)

5

2

=

(

)

5

3

=

5!

2! 3!

=

5 ∙ 4

2 ∙ 1

=

10

and so on.

Now we consider the following powers of
  a + b:

(

)

0

0

a0 + b0

(a + b)0 = 1 =

(a + b)1 = a + b =

(

)

1

0

(

)

1

1

a0b1

a1b0 +

(a + b)2 = a2 + 2ab + b2

(

)

2

1

a1b1 +

(

)

2

2

a0b2

=

(

)

2

0

a2b0 +

(a + b)3 = a3 + 3a2b +
 3ab
2 + b3

=

(

)

3

0

a3b0 +

(

)

3

1

a2b1 +

(

)

3

2

a1b2 +

(

)

3

3

a0b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

(

)

4

1

a3b1 +

(

)

4

2

a2b2 +

=

(

)

4

0

a4b0 +

(

)

4

3

a1b3 +

(

)

4

4

a0b4

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

(

)

5

1

a4b1 +

(

)

5

2

a3b2 +

=

(

)

5

0

a5b0 +

(

)

5

3

a2b3 +

(

)

5

4

a1b4 +

a0b5

(

)

5

5

For each (a + b)n expansion:

  1. There are (n + 1) terms,
  2. The exponents of a decrease by 1 and the exponents of b increase by 1 for each subsequent term,
  3. The sum of the exponents of a and b in each term is n,
  4. The coefficient of an - i bi is:

(

n

i

)

This pattern continues for larger values of n and is the Binomial Expansion Formula:

(

)

n

0

anb0 +

(a + b)n =

(

)

n

1

an – 1 b1 +

(

)

n

2

an – 2 b2

. . .

+

+

(

)

n

n – 2

a2bn – 2

(

)

n

n

a0bn

a1bn – 1

)

(

n

n – 1

+

(

)

n

i

are binomial coefficients.

The coefficients

The Binomial Expansion Formula can be stated using Σ notation:

(a + b)n =

)

an – i bi

n

Σ

i = 0

(

n

i

The primary interest of advanced factorial math is to understand a systematic way to determine the positive integral powers of a binomial, a + b. The following computations are important to understand expanding powers of a + b: