Counting II 3, 7 3, 2 3, 9 7, 2 7, 9 2, 9

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1 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 di eret 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 di eret 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 di eret 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!.

2 ! 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 di eret 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 di eret 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 i a small library. You wat to chec out boos. The library has 8 di eret boos to choose from. There are 8 di eret collectios of three boos that you could chec out. 8 8!!(8 )! 8(8)(8)(8!)! 8!, 8,, 8(8)(8)!, 9, 9, 9 * * * * * * * * * * * * *

3 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 di eret 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, a..a. objects, from a set is. That meas. Now we ca fill i the values for ad ito Pascal s triagle.

4 There are di eret ways to choose object from a set that has objects. Thus,. Similarly, there are exactly di eret 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

5 asetof objects is the same as leavig. Therefore, This formula tells us that the rows i Pascal s triagle will read the same left-to-right as they will right-to-left. You ca see that i the triagle o the ext page. What we saw earlier i the formula 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 di eret rocs: big red bric, ad di eret little blue marbles. How may di eret 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 di eret ways we could choose marbles from the total umber of littlebluemarbles. Thus,thereare di eret 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,adtheumberofdi eret 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.

6 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. * * * * * * * * * * * * *

7 Biomial Theorem For x, y R ad N, (x + y) X i x i The two lies above are what is called the Biomial Theorem. It gives you a easy way to fid powers of sums. Example. We ca use the Biomial Theorem ad Pascal s triagle to write out the product (x + y). The Biomial Theorem states that X (x + y) x i y i i i x y + x y + x + x y + xy +,, i y i 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 coe ciets. Because of the Biomial Theorem, umbers of the form are called biomial coe ciets. Example. For ay N, we ca let x ad y both be. The umber raised to ay power equals, so x i ady i. Also, x + y. So writig the Biomial Theorem whe x ady tellsusthat X i Notice that the equatio o the above lie is exactly i +

8 It says that is the sum of all of the umbers i the row of Pascal s triagle that begis with. For example, is the sum of the umbers,,,ad. I other words, , as you ca see. 8 is what you d get if you added the umbers i the last row of Pascal s triagle as it s writte o page. * * * * * * * * * * * * * 8

9 Exercises For #-, use that the umber of ways to choose objects from a set of objects is!!( )!.) A small coutry has just udergoe a revolutio ad they commissio you to desig a ew flag for them. They wat exactly colors used i their flag, ad they have give you colors of cloth that you are allowed to use. How may di eret 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 di eret types of bagels. If the shop has di eret ids of bagels to choose betwee, the how may di eret variety pacs does a customer have to choose from?.) To play the lottery you have to select out of 9 umbers. How may di eret ids of lottery ticets ca you purchase? For #-8, use the Biomial Theorem ad Pascal s triagle. Your aswers for #- should have a similar form to the aswer give i the first example o page..) Write out the product (x + y)..) Write out the product (x + y)..) Fid (x +). 8.) Fid (x ). 9.) The formula i defies a sequece. Is it arithmetic or geometric?.) Compute the series X i 9 i