1. Lecture 10 To define the Binomial Theorem
To understand the relationship between Pascal’s Triangle and the Binomial Theorem
To apply the Pigeonhole Principle in counting

2. Binomial Theorem and so on (x + y)1 = x1 + y1 (x + y)2 = x2 + 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy2 + y3
and so on

3. The Binomial Theorem
For every non-negative integer n,

4. Example (Binomial Theorem)
Find the coefficient of a5b4 in the expansion (a + b)9.

5. Example (Binomial Theorem)
The binomial theorem has many applications. For example, it can be used to produce approximations.
Estimate (2.01)3 to 2 decimal places.

6. Pascal’s Triangle 1

7. Pascal’s Triangle
Every number below is the sum of the two numbers closest to it in the row above, one number being above and just to the left while the other number is above and just to the right.
The rows of Pascal’s triangle appear in a wide variety of situations.

8. Pascal’s Triangle and the Binomial Theorem
What would be the next row in the Pascal’s Triangle in the earlier slide?
The next row would be:
Because of the relationship between Pascal’s triangle and the binomial theorem, from this new row we can deduce that
(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5

9. Weight of a Word
Consider the set of all binary strings (or “words”) with 3 bits.
In Coding Theory and Language Theory it’s often important to identify the different weights among binary words with some fixed number of bits.
The weight w(x) of a binary word x is the number of times that 1 appears in the word.
Example: w(101) = 2

10. Weight of a Word
Among 3-bit words, what weights occur and how many different words are there having any specified weight?
3 bits
weight
count
000 1
100 3
010
001
101 2
110
011
111

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Weight of a Word
Of course, the same question can be asked about 1-bit, 2-bit, 4-bit words, etc.
Among n-bit words, how many have weight r (where 0 ≤ r ≤ n)? The answer is nCr – the number of ways to select the r positions where 1 is to occur.
2 bits
weight
count 00 1 10 2 01 11
1 bit
weight
count 1
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Weight of a Word
When n = 1 we get: 1, 1
When n = 2 we get: 1, 2, 1
When n = 3 we get: 1, 3, 3, 1
Leaving out commas, we get several familiar rows of numbers. If we insert an extra row at the top, consisting of a single 1, we get the famous pattern known as Pascal’s Triangle.
RMIT University; Taylor's College

13. Binomial Coefficients
In general, the set of all n-bit words will have nC0 (=1) word of weight 0, nC1 words of weight 1, nC2 words of weight 2, and so on.
It will have nCn-1 (= nC1 = n) words of weight (n – 1), and nCn ( = nC0 = 1) word of weight n
These numbers
nC0, nC1, nC2, …, nCn
are called binomial coefficients.
They appear in the following well known theorem.
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14. Binomial Theorem: Corollary
Here’s a useful corollary to the binomial theorem. For every non-negative integer n,

15. Binomial Coefficients and the Weight of Words
Let’s look at the lists of binomial coefficients.
Now if we look back at the question of the weights of the 3-bit binary words, we can see how the numbers arise.
There are 23 = 8 words that have 3 bits. Of those:
3C0 (=1) has weight 0
3C1 (=3) have weight 1
3C2 (=3C1 = 3) have weight 2
3C3 (=3C0 = 1) has weight 3
Now we move on, and look at an apparently obvious and yet exceedingly useful principle which is often applied in situations where we want to count things.
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16. The Pigeonhole Principle
If (n + 1) or more pigeons are put into n holes, at least two pigeons must be in the same hole.
Also known as the Dirichlet Drawer Principle or the Shoe Box Principle
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Example
Among 13 people there are two who have their birthdays in the same month.
There are n married couples. How many of the 2n people must be selected in order to guarantee that one has selected a married couple?
In any group of n people there are at least two persons having the same number friends. (It is assumed that if a person x is a friend of y then y is also a friend of x.)
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