Algebra/Chapter 10/Binomial Theorem

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Factorials

The notation ' $n!$ ' is defined as n factorial.

n! = n \times (n - 1) \times (n - 2) \times (n - 3) \times \dots \times 3 \times 2 \times 1

0 factorial is equal to 1.

0! = 1

Proof of 0 factorial = 1

n! = n \times (n - 1)!

When n = 1,

1! = 1 \times (1 - 1)!

1 = 1 \times 0!

And thus,

0! = 1

The Binomial Theorem

The binomial thereom gives the coefficients of the polynomial

(x + y)^{n}

.

We may consider without loss of generality the polynomial, of order n, of a single variable z. Assuming $x \neq 0$ set z = y / x

(x + y)^{n} = x^{n} (1 + z)^{n}

.

The expansion coefficients of $(1 + z)^{n}$ are known as the binomial coefficients, and are denoted

(1 + z)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) z^{k}

.

Noting that

(x + y)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{n - k} y^{k}

is symmetric in x and y, the identity

(\binom{n}{n - k}) = (\binom{n}{k})

may be shown by replacing k by n - k and reversing the order of summation.

A recursive relationship between the $(\binom{n}{k})$ may be established by considering

(1 + z)^{n + 1} = (1 + z) (1 + z)^{n} = \sum_{k = 0}^{n + 1} (\binom{n + 1}{k}) z^{k} = (1 + z) \sum_{k = 0}^{n} (\binom{n}{k}) z^{k}

or

\sum_{k = 0}^{n + 1} (\binom{n + 1}{k}) z^{k} = \sum_{k = 0}^{n} (\binom{n}{k}) z^{k} + \sum_{k = 0}^{n} (\binom{n}{k}) z^{k + 1} = \sum_{k = 0}^{n} (\binom{n}{k}) z^{k} + \sum_{k = 1}^{n + 1} (\binom{n}{k - 1}) z^{k}

.

Since this must hold for all values of z, the coefficients of $z^{k}$ on both sides of the equation must be equal

(\binom{n + 1}{k}) = (\binom{n}{k}) + (\binom{n}{k - 1})

for k ranging from 1 through n, and

(\binom{n + 1}{n + 1}) = (\binom{n}{n}) = \frac{n!}{(n - n)! n!} = \frac{n!}{n!} = 1

(\binom{n + 1}{0}) = (\binom{n}{0}) = \frac{n!}{(n - 0)! 0!} = \frac{n!}{n!} = 1

.

Pascal's Triangle is a schematic representation of the above recursion relation ...

Show

(\binom{n}{k}) = \frac{n!}{k! (n - k)!}

(proof by induction on n).

A useful identity results by setting $z = 1$

\sum_{k = 0}^{n} (\binom{n}{k}) = 2^{n}

.

The visual way to do the binomial theorem

Template:Wikipedia (this section is from difference triangles)

Lets look at the results for (x+1)ⁿ where n ranges from 0 to 3.

(x+1)⁰ =          1x⁰           =                1         
(x+1)¹ =        1x¹+1x⁰         =              1   1
(x+1)² =      1x²+2x¹+1x⁰       =            1   2   1
(x+1)³ =    1x³+3x²+3x¹+1x⁰     =          1   3   3   1

This new triangle is Pascal’s Triangle.

It follows a counting method different from difference triangles.

The sum of the x-th number in the n-th difference  and 
the (x+1)-th number in the n-th difference yields the
(x+1)-th number in the (n-1)-th difference.

It would take a lot of adding if we were to use the difference triangles in the X-gon to compute (x+1)¹⁰. However, using the Pascal’s Triangle which we have derived from it, the task becomes much simpler. Let’s expand Pascal’s Triangle.

(x+1)⁰                                    1
(x+1)¹                                  1   1
(x+1)²                                1   2   1
(x+1)³                             1    3   3    1
(x+1)⁴                           1    4   6   4    1
(x+1)⁵                         1   5   10   10   5   1
(x+1)⁶                      1   6   15   20   15   6   1
(x+1)⁷                   1   7   21   35   35   21   7    1
(x+1)⁸                1   8   28   56   70   56   28   8    1
(x+1)⁹              1   9   36   84   126  126  84   36   9    1
(x+1)¹⁰          1   10  45   120  210  252  210  120  45   10    1

The final line of the triangle tells us that

(x+1)¹⁰ = 1x¹⁰ + 10x⁹ + 45x⁸ + 120x⁷ + 210x⁶ + 252x⁵ + 210x⁴ + 120x³ + 45x² + 10x¹ + 1x⁰.

Example Problems

Practice Problems

<quiz display=simple points=1/1> {Problem 1: Find the following without using a pencil and paper:}

{ $9 9^{2}$ |type="{}"} { 9801_7 }

{ $99 8^{2}$ |type="{}"} { 996004_7 }

{ $10 1^{3}$ |type="{}"} { 1030301_7 } </quiz>

Problem 2: If 3! * 5! * 7! = n!, what is n?

Algebra/Chapter 10/Binomial Theorem

Contents

Factorials

The Binomial Theorem

The visual way to do the binomial theorem

Example Problems

Practice Problems

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Algebra/Chapter 10/Binomial Theorem

Factorials

The Binomial Theorem

The visual way to do the binomial theorem

Example Problems

Practice Problems

Navigation menu

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