The Binomial Theorem

Advanced Higher Notes (Unit 1)

The Binomial Theorem

The Binomial Theorem

Prerequisites: Cancelling fractions; summation notation; rules of indices.

Maths Applications: Proving trig. identities using complex numbers; probability.

Real-World Applications: Counting problems; Hardy-Weinberg Formula (biology).

Factorials and Binomial Coefficients

In a race with 3 people, in how many ways can the runners finish ? There are 3 possibilities for the first place; for each of these 3 possibilities, there are 2 possibilities for the remaining 2 places; for each of these 2 possibilities, there is only 1 possibility for the final place. So, there are 3 ? 2 ? 1 = 6 ways the runners can finish.

Definition:

For n ? , the factorial of n (aka n factorial or factorial n) is,

def

n ! = n ? (n - 1) ? (n - 2) ? ... ? 3 ? 2 ? 1

def

Note that 1! = 1 and the convention 0! = 1 is made.

In how many ways can a child pick 3 crayons from a selection of 5 coloured crayons (all different colours) ? There are 5 ? 4 ? 3 = 60 ways of choosing 3 crayons, if the order in which they are taken matters. However, the child isn't interested in which order they're picked, so this answer of 60 is too big. Whichever 3 colours are picked, there are 3 ! = 6 ways of doing so. Therefore, the value of 60 is 6 times too large; hence, the actual number of ways of choosing 3 crayons from 5 without worrying about the order is 60 ? 6 = 10.

The order matters in a permutation, as opposed to a combination.

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Advanced Higher Notes (Unit 1)

The Binomial Theorem

Definition:

The number of ways of choosing r objects from n without taking into account the order (aka n choose r or the number of combinations of r objects from n) is given by the binomial coefficient nCr defined by,

nC r

n

def

=

r

n! r ! (n - r )!

Evaluating a Binomial Coefficient Without a Calculator

Example 1

7C 4

=

7! 4! (7 - 4)!

= 7?6?5?4?3?2?1 (4 ? 3 ? 2 ? 1) ? (3 ? 2 ? 1)

= 7?6?5 3?2?1

= 35

Properties of Binomial Coefficients

?

nr

=

n

n -

r

?

r

n -

1

+

nr

=

n

+ r

1

(Khayyam-Pascal Identity)

Example 2

Solve the equation

n

n -

2

= 15.

n! 2! (n - 2)!

= 15

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Advanced Higher Notes (Unit 1)

The Binomial Theorem

n (n - 1) (n - 2)! 2 (n - 2)!

= 15

n (n - 1) 2

= 15

n 2 - n = 30

n 2 - n - 30 = 0

(n - 6) (n + 5) = 0

Hence, n = 6 or n = - 5. However, as n cannot be negative, n = 6.

Example 3

Show that n

+

1

3

-

n

3

=

n

.

2

A very simple proof can be obtained by replacing r with 2 in the Khayyam-Pascal Identity. However a more ` get your fingers dirty ' method will be given, which illustrates some general techniques when manipulating binomial coefficients.

Starting with the LHS,

n

+

1

3

-

n

3

=

(n + 1)! 3! (n - 2)!

-

n! 3! (n -

3) !

(n + 1) n ! = 3! (n - 2) (n - 3)!

-

n! 3! (n - 3)!

=

n! 3! (n -

3) !

n n

+1 -2

-

1

=

n! 3! (n -

3) !

n

+1 n

-n + -2

2

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Advanced Higher Notes (Unit 1)

The Binomial Theorem

=

n! 3! (n -

3) !

n

3 -

2

n! 3 = 3 . 2! (n - 3)! (n - 2)

n! = 2! (n - 2)!

=

n

2

which equals the RHS.

Pascal's Triangle

Pascal's triangle - also known as Pingala's triangle, Khayyam's triangle, Yang Hui's triangle and Tartaglia's triangle, after mathematicians who discovered or studied the triangle before Pascal ? is the following infinite arrangement of evaluated binomial coefficients:

1 1 1 1 2 1 13 31 1 4 6 4 1 1 5 10 10 5 1

The top row consisting of the single entry 1 is the 0th row. Each number not on an edge of the triangle is obtained by adding the 2 numbers in the previous row and just to the right and left of that entry (this is the Khayyam-Pascal identity).

M Patel (April 2012)

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St. Machar Academy

Advanced Higher Notes (Unit 1)

The Binomial Theorem

The Binomial Theorem

Taking powers of a binomial can be achieved via the following theorem.

Theorem (Binomial Theorem):

For whole numbers r and n,

(x + y) n

n

=

nCr x n -r y r

r =0

Written out fully, the RHS is called the binomial expansion of (x + y) n .

Using the first property of the binomial coefficients and a little relabelling, the Binomial Theorem can be written slightly differently.

Corollary:

(x + y) n

n

=

nCr x r y n -r

r =0

Note that there are n + 1 terms in any binomial expansion.

Expanding a Binomial

Example 4

Expand (x + y) 5 .

5

(x + y) 5 =

5 r

x

5 -r

yr

r =0

=

5

x5

y 0

0

+

5

x4

y 1

1

+

5

x3

y 2

2

+

5

x2

y 3

3

+

5

x1

y 4

4

+

5

x

0

y 5

5

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