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Lecture 10

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. Binomial Theorem. ( x + y ) 1 = x 1 + y 1 ( x + y ) 2 = x 2 + 2 xy + y 2

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Lecture 10

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  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 • (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

  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: 1 5 10 10 5 1 • 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?

  11. 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. RMIT University; Taylor's College

  12. 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. RMIT University; Taylor's College

  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. RMIT University; Taylor's College

  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 RMIT University; Taylor's College

  17. 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.) RMIT University; Taylor's College

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