Coutig II Sometimes we will wat to choose objects from a set of objects, ad we wo t be iterested i orderig them For example, if you are leavig for vacatio ad you wat to pac your suitcase with three of the seve pairs of shorts that you ow, the it does t matter i which order you pac the shorts All that matters is which three pairs you pac choose The umber of differet ways that objects ca be chose from a set of objects (whe order does t matter) is called choose It is writte i symbol form as Examples There are four differet ways that oe letter ca be chose from the set of four letters {e, f, g, a} Oe way is to choose the letter e Alteratively, you could also choose the letter f, or the letter g, or the letter a Sice there are optios for choosig oe object from a set of objects, we have Below is a list of all the possible ways that umbers ca be chose from the set of four umbers {,,, 9} There are six differet ways Thus,,,, 9,, 9, 9 Geeral formula To say that we are choosig ad orderig objects from a set of objects is to say that we are performig separate tass First is the tas of choosig objects from the set of objects, ad the umber of ways to perform that tas is Secod is the tas of orderig the objects after we ve chose them There are! ways to order objects Let s repeat that To choose ad order objects: First, choose the objects, the order the objects you chose Optios multiply, so the total umber of ways that we ca choose ad order objects from a set of objects is!
! We saw i the previous chapter that there are exactly ( )! ways to choose ad order objects from a set of objects Therefore,!! ( )! Dividig the previous equatio by!: Examples!!( )! There are differet ways to choose which of the pairs of shorts that you will tae o your vacatio!!( )! ()()(!)!! ()() () How may card poer hads are there if you play with a stadard dec of cards? You re coutig the umber of differet collectios of cards that ca be tae from a set of cards This umber is!!( )! ()()(9)(8)(!)!! ()()(9)(8) ()()(9)(8)!, 98, 9 You are at a DVD retal store You wat to ret DVDs The store has 8 differet DVDs to choose from There are 8 differet collectios of three movies that you could ret 8 8!!(8 )! 8(8)(8)(8!)! 8!, 8,, 8(8)(8)!, 9, 9, 9 * * * * * * * * * * * * *
Pascal s Triagle We ca arrage the umbers ito a triagle I each row, the top umber of is the same The bottom umber of is the same i each upward slatig diagoal The triagle cotiues o forever The first 8 rows are show above This is called Pascal s triagle It is amed after a Frech mathematicia who discovered it It had bee discovered outside of Europe ceturies earlier by Chiese mathematicias Moder mathematics bega i Europe, so its traditios ad stories ted to promote the exploits of Europeas over others Some values of to start with is the umber of differet ways you ca select objects from a set of objects There is oly oe way to tae everythig you just tae everythig so Similarly, there is oly oe way to tae othig from a set just tae othig, that s your oly optio The umber of ways you ca select othig, aa objects, from a set is That meas Now we ca fill i the values for ad ito Pascal s triagle 8
There are differet ways to choose object from a set that has objects Thus, Similarly, there are exactly differet optios for choosig objects from a set of objects To see this, otice that decidig which of the objects that you will tae from a set of objects is the same as decidig which object you will leave behid So the umber of ways you ca tae objectsisthesameastheumberofwaysyoucaleaveobject That is to say, Hece, Oce we fill i this ew iformatio o Pascal s triagle it loos lie Before movig o, let s loo bac at the last rule that we foud: The argumet we gave there geeralizes, i that taig objects from 9
asetof objects is the same as leavig Therefore, What we saw earlier i the form was the special case of the formula whe Add the two umbers above to get the umber below Suppose that you have a set of differet rocs: big red bric, ad differet little blue marbles How may differet ways are there to choose +rocsfromthesetof rocs? Ay collectio of + rocs either icludes the big red bric, or it does t Let s first loo at those collectios of +objectsthatdo cotai a big red bric Oe of the +objectswewillchooseisabigredbric That s agive Thatmeasthatallwehavetodoisdecidewhich of the little blue marbles we wat to choose alog with the big red bric to mae up our collectio of +objects Thereare differet ways we could choose marbles from the total umber of littlebluemarbles Thus,thereare differet ways we could choose a set of +objectsfromoursetof rocs if we ow that oe of the objects we will choose is a big red bric Now let s loo at those collectios of + objects that do t cotai the big red bric The all +objectsthatwewillchoosearelittlebluemarbles There are littlebluemarbles,adtheumberofdifferet ways we could choose +ofthe littlebluemarblesis + Ay collectio of + rocs either icludes the big red bric, or it does t So to fid the umber of ways that we could choose +objects, wejust have to add the umber of possibilities that cotai a big red bric, to the umber of possibilities that do t cotai a big red bric That formula is + + + If ad,thetheaboveformulasaysthat + Looig at Pascal s triagle, you ll see that ad are the two umbers that are just above the umber Chage the values of ad ad chec that the above formula always idicates that to fid a umber i Pascal s triagle, just sum the two umbers that are directly above it
For example, + + ad + + We ow the atural umbers that are at the very tip of Pascal s triagle To fid the rest of the umbers i Pascal s triagle, we ca let that owledge tricle dow the triagle usig this latest formula that ay umber i the triagle is the sum of the two umbers above it Notice that Pascal s triagle is the same if you read it left-to-right, or right-to-left Covice yourself that this is a cosequece of the formula * * * * * * * * * * * * * Biomial Theorem For x, y R ad N, (x + y) i x i y i i
Example We ca use the biomial theorem ad Pascal s triagle to write out the product (x + y) The biomial theorem states that (x + y) x i y i i i x y + x y + x + x y + xy + x y + y x y The umbers,,,ad mae up the fourth row of Pascal s triagle, ad we ca see from the triagle that they equal,,, ad respectively Therefore, (x + y) x +x y +xy + y Biomial coefficiets Because of the biomial theorem, umbers of the form are called biomial coefficiets * * * * * * * * * * * * *
Exercises ) A small coutry has just udergoe a revolutio ad they commissio you to desig a ew flag for them The wat exactly colors used i their flag, ad they have give you colors of cloth that you are allowed to use How may differet color combiatios do you have to decide betwee? ) A sports team is sellig seaso ticet plas They have home games i a seaso, ad they allow people to purchase ticets for ay combiatio of home games How may ways are there to choose a collectio of home games? ) A bagel shop ass customers to create a Baer s Doze Variety Pac by choosig differet types of bagels If the shop has differet ids of bagels to choose betwee, the how may differet variety pacs does a customer have to choose from? ) To play the lottery you have to select out of 9 umbers How may differet ids of lottery ticets ca you purchase? ) Use the Biomial Theorem ad Pascal s triagle to write out the product (x + y) (Your aswer should have a similar form to the example that was doe before the exercises) ) Use the Biomial Theorem ad Pascal s triagle to write out the product (x + y) ) Fid (x +) usig the Biomial Theorem 8) Fid (x ) usig the Biomial Theorem