1. n! = n. tion. For example, (n+1)! working with factorials. = (n+1) n (n 1) 2 1

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1 Biomial Coefficiets ad Permutatios Mii-lecture The followig pages discuss a few special iteger coutig fuctios You may have see some of these before i a basic probability class or elsewhere, but perhaps you have t used them i full geerality These fuctios are icredibly useful ad they are iterestig to study i their ow right We will use each of these fuctios from time to time throughout the quarter Let s start with some defiitios: 1! i1 i 1 1 umber of ways to arrage permute the etire set [] {1,,, } i some order umber of ways to defie a bijective fuctio f from [] to [] factorial By covetio, we also defie 0! 1 Examples: 4! ! , so fractios ivolvig factorials ofte have a lot of cacella This is useful to remember whe ! Note: ! tio For example, +1! worig with factorials! P, umber of ways to arrage permute elemets from the set [] {1,,, } umber of ways to defie a ijective fuctio from [] to [] Examples: P 6, P 6, I other words, P, is just lie the factorial except you stop after multiplyig umbers startig at C, umber of ways to choose elemets from the set [] {1,,, } order does t matter choose The formulas for this are somewhat less direct, they all use the two previous fuctios To itemize the ways to choose items sometimes taes a bit loger Examples will be give below 0 By covetio, we also defie 1 ad if > or < 0 we defie 0 0 O the ext page, we explore coectios betwee these fuctios to get a formula for

2 The Coectio Betwee P, ad C, 1 Cosider the set [4] {1,,, 4} ad let Ways to choose elemets from [4] Ways to arrage elemets from [4] {1, } 1,,,1 {1, } 1,,,1 {1, 4} 1,4, 4,1 {, },,, {, 4},4, 4, {, 4},4, 4, 4 a Thus, the umber of ways to choose from {1,,, 4} is C 4, 6 b The umber of ways to arrage elemets from {1,,, 4} is 1 which matches up with the formula: P 4, because + 1 c NOTE THAT P 4, C 4, Cosider the set [] {1,,, 4, } ad let Ways to choose elemets from [] Ways to arrage elemets from [] {1,, } 1,,, 1,,,,1,,,,1,,1,,,,1 {1,, 4} 1,,4, 1,4,,,1,4,,4,1, 4,1,, 4,,1 {1,, } 1,,, 1,,,,1,,,,1,,1,,,,1 {1,, 4} 1,,4, 1,4,,,1,4,,4,1, 4,1,, 4,,1 {1,, } 1,,, 1,,,,1,,,,1,,1,,,,1 {1, 4, } 1,4,, 1,,4, 4,1,, 4,,1,,1,4,,4,1 {,, 4},,4,,4,,,,4,,4,, 4,,, 4,, {,, },,,,,,,,,,,,,,,,, {, 4, },4,,,,4, 4,,, 4,,,,,4,,4, {, 4, },4,,,,4, 4,,, 4,,,,,4,,4, a Thus, the umber of was to choose from {1,,, 4, } is C, 10 b The umber of ways to arrage elemets from {1,,, 4, } is 60 which matches up with the formula: P, because + 1 c NOTE THAT P 6, 6 C 6, AND 6! 4 I summary, we see that P 4, 4! I geeral, P,!C,, thus Therefore, whe we solve for 1 + 1! we get 1 + 1!!C 4, ad P, 4! which is ofte writte as!c,!!!

3 Here are a few examples where we use the formula from the previous page:!! Note that!!!! 1 1 coicidece, is always true 7 7!!7! 7!!4! , this is ot a 6! Note: , so fractios ivolvig factorials ofte have a lot of cacellatio 4! For example, +1! This is useful to remember whe worig! with factorials Pascal s Idetity The umbers have some ice properties if you list them out Most otably is the recurrece formula attributed to Pascal Here the first several values of are give is the row ad is the colum See if you ca spot a relatioship betwee the rows ad colums, the aswer immediately follows the table so try to fid the patter before you read o The relatioship is that each etry is the sum of the two etries immediately above ad immediately 4 4 above ad to the left For example, + This relatioship is give i the followig theorem Pascal s Idetity: For all, Z such that 0 <, The Biomial Theorem The mai reaso we are discussig these fuctios is because of their appearace i the biomial theorem Whe we expad out a biomial to various powers a patter appears For example I ecourage you to chec these expasios by doig the multiplicatio out the log way x + y x + xy + y x + y x + x y + xy + y

4 x + y 4 x 4 + 4x y + 6x y + 4xy + y 4 There are familiar umbers I fact the coefficiet of x y is the expasio of x + y is always, for this reaso the umbers as ofte referred to as biomial coefficiets So we have the followig theorem: The Biomial Theorem: For all Z ad for all x, y R, 0 x y x + y x 1 y x y x y x 1 y 1 + x 0 y For example, x + y x 6 y x 6 + 6x y + 1x 4 y + 0x y + 1x y 4 + 6xy + y 6

5 Practice Problems 1 Prove that Hit: The proof will be short, you ca either use a combiatorial argumet for why they would give the same umber or use the ow formula for biomial coefficiets + 1 Prove that if ad are both eve, the is eve So the umber 1 below two eve umbers i Pascal s triagle is also eve a Usig the biomial theorem, tell me what eeds to go i place of the questio mars i 1 + a + a + a + + a 1 + a???? b Use the formula from part a, to compute the value give by 0 1 c What is 11 4? How ca you use the formula i part a to easily compute this value by had?