The Binomial Theorem
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1 The Biomial Theorem Lecture 47 Sectio 9.7 Robb T. Koether Hampde-Sydey College Thu, Apr 8, 03 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 / 7
2 Combiatios Pascal s Triagle 3 The Biomial Theorem 4 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 / 7
3 Outlie Combiatios Pascal s Triagle 3 The Biomial Theorem 4 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 3 / 7
4 Combiatios Theorem Let ad r be oegative itegers with r. The ( ( =. r r Proof. To choose which r elemets to iclude i the subset is the same as choosig which r elemets ot to iclude. Thus, ( ( r = r. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 4 / 7
5 A Recurrece Relatio Theorem Let ad r be positive itegers with r <. The ( ( ( = +. r r r Proof. Let A be a set of elemets ad let x A. Divide the subsets of size r ito two groups: ( Those that cotai x. ( Those that do ot cotai x. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 5 / 7
6 A Recurrece Relatio Proof. How may subsets are i group (? Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 6 / 7
7 A Recurrece Relatio Proof. How may subsets are i group (? If we remove x from each, we have all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 6 / 7
8 A Recurrece Relatio Proof. How may subsets are i group (? If we remove x from each, we have all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. How may subsets are i group (? Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 6 / 7
9 A Recurrece Relatio Proof. How may subsets are i group (? If we remove x from each, we have all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. How may subsets are i group (? The elemet x is i oe of them, so if we remove x from A, these subsets are all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 6 / 7
10 A Recurrece Relatio Proof. How may subsets are i group (? If we remove x from each, we have all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. How may subsets are i group (? The elemet x is i oe of them, so if we remove x from A, these subsets are all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( such subsets. Thus, ( r ( = ( + r r r. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 6 / 7
11 Outlie Combiatios Pascal s Triagle 3 The Biomial Theorem 4 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 7 / 7
12 Pascal s Triagle The equatio ( = r ( r + ( r allows us to computer ( r recursively. The recursio eds with the boudary cases ( 0 = ad ( =. This is the basis of Pascal s Triagle. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 8 / 7
13 Pascal s Triagle 3 r Iitialize the boudary to Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 9 / 7
14 Pascal s Triagle 3 r Compute ( 3 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 9 / 7
15 Pascal s Triagle 3 r Compute ( ( 4 ad 4 3 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 9 / 7
16 Pascal s Triagle 3 r Compute ( ( 5, 5 ( 3, ad 5 4 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 9 / 7
17 Outlie Combiatios Pascal s Triagle 3 The Biomial Theorem 4 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 0 / 7
18 The Biomial Theorem Theorem Let be a oegative iteger ad let a ad b be ay real umbers. The ( (a + b = a + a b + ( = a i b i. i i=0 ( ( a b + + ab + b Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 / 7
19 The Biomial Theorem Proof. The proof is by iductio o. Whe = 0, we have (a + b 0 = ad 0 i=0 ( a i b i = i =. ( 0 a 0 0 b 0 0 Therefore, the statemet is true whe = 0. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 / 7
20 The Biomial Theorem Proof. Suppose that the statemet is true for some iteger k where k 0. The (a + b = (a + b(a + b ( = (a + b a i b i i i=0 ( ( = a i b i + a i b i+ i i i=0 i=0 ( ( = a i b i + a i b i i i i=0 i= Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 3 / 7
21 The Biomial Theorem Proof. ( = a + a i b i + i i= i= [ ( = a + + i i= ( = a + a i b i + b i = i= ( a i b i. i i=0 ( a i b i + b i ( ] a i b i + b i Therefore, the statemet is true whe = k +, ad so it is true for all 0. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 4 / 7
22 Examples Expad (a + b 5. Expad (a b 5. Expad (a + b 5. Show that ( ( 0 + ( + ( + + =. Show that ( ( 0 ( + ( ± = 0. What is the value of ( ( ( + + (? What is the value of ( ( 0 + ( ± (? Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 5 / 7
23 Outlie Combiatios Pascal s Triagle 3 The Biomial Theorem 4 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 6 / 7
24 Assigmet Assigmet Read Sectios 9.7, pages Exercises 0,,, 6, 8,, 6, 30, 3, 39, 4, page 603. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 7 / 7
The Binomial Theorem
The Biomial Theorem Lecture 47 Sectio 9.7 Robb T. Koether Hampde-Sydey College Fri, Apr 8, 204 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, 204 / 25 Combiatios 2 Pascal s Triagle
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