Combinatorics and Newton s theorem

Transcription

1 INTRODUCTION TO MATHEMATICAL REASONING Key Ideas Worksheet 5 Combiatorics ad Newto s theorem This week we are goig to explore Newto s biomial expasio theorem. This is a very useful tool i aalysis, but it also offers us a opportuity to explore a iterestig coectio with the combiatorics of fiite sets. Newto s theorem is a formula for the expasio of a expressio of the form (x+y), for ay atural umber. Oce oe has the formula, the a proof by iductio ca show that the formula is true. However, the coectio to combiatorics actually explais why the formula holds. So, as usual, we will overproof our statemet. Let us start with some defiitios. Defiitio. For ay atural umber, we defie! (read factorial ) to be the product of all atural umbers from to (icludig ). So for example!,!,! 6,!, 5!,... There are good reasos to defie!, eve though we will ot get ito them right ow. Defiitio. For ay two atural umbers, k, with k <, we defie ( ) k (read choose k ) to be: ( )! k k!(! So for example ( ( ) 6, ( ) ). Sice we defied!, we ca also cosistetly defie ( ( ) ). With these two defiitios i place, we are ready to state Newto s biomial expasio theorem. Theorem. For ay atural umber, we have: ( ) (x + y) x k y k. k k Let us see this theorem i actio. If you wat to expad (x + y) 6, you ca immediately write: (x + y) 6 x 6 + 6x 5 y + 5x y + x y + 5x y + 6xy 5 + y 6, ( because, i this case, we have: 6 6 6, , 6 6 ) 5, ( 6 ). Let us take a side-step for a secod, ad observe that the umber! couts the umber of ways you ca order the elemets of a set with elemets.

2 Theorem. Let X {x,..., x } be a set with elemets. There are! distict ways of orderig the elemets of X. Proof. We have choices for what elemet to put i the first positio. Oce we have chose a elemet for first positio, we have choices for what elemet to put i secod positio. Oce we have chose the secod elemet, we have choices for what elemet to put i third positio. This patter cotiues util we have oly oe elemet left, which must be put i the last positio. All together we have ( ) ( )... choices, which is precisely the defiitio of!. Groupwork Let us start by givig the umber ( a combiatorial meaig. Problem. If X is a set with elemets, show that ( is the umber of subsets of X which cotai k elemets. Questio. After solvig Problem, aswer the followig:. Why is ( k?. Are the defiitios ( ) ( ) cosistet with Problem? Problem. Show that the followig idetity holds: ( ) + k k k Show it i two differet ways:. Use the algebraic defiitio ad do some algebra.. Try to subdivide the set of subsets of X with k elemets ito two subsets, oe of cardiality ( k, ad the other of cardiality k ). Now let us tackle the proof of Newto s biomial theorem. To warm up, let us do the followig exercise. Problem. Write dow (x + y)(x + y)(x + y) i three differet colors, oe for each biomial. Now expad the above expressio keepig track of the colors. For example, istead of writig x, do write xxx. Subdivide the terms ito four groups, correspodig to the four moomials x, x y, xy, y that you obtai whe you forget the colors. Ca you relate the umber of elemets i each group with a umber that has to do with the set of colors X {black, blue, red}? Here comes the mai problem i this worksheet, so make sure you sped a fair amout of time o this ad uderstad it very well. Problem. Geeralize the idea from the previous problem to give a proof of Theorem.

3 Let us coclude this groupwork with some fu exploratio. Problem 5 (Pascal s triagle). Pascal s triagle is a ifiite triagle of umber that cotais all umbers of the form (, as show i the followig figure: ( ) ( ) ( ) ( ) ( ) ( ) ( 6 5 6)

4 Pluggig i the umbers, you get Spot as may iterestig umerical patters as you ca i Pascal s triagle. The, try to explai them! Suday Homework Exercise. Give a proof by iductio of Newto s Theorem (Theorem ). Exercise. Use Newto s theorem to show the followig facts: The sum of the umbers i each row of Pascal s triagle is a power of. The alteratig sum of the umbers i each row of Pascal s triagle is equal to. Exercise. Give a atural umber, ad three umbers k, k, k such that k +k +k, what is the umber of ways you ca subdivide X ito three disjoit subsets amed U, U, U, where the cardiality of U is k, the cardiality of U is k ad the cardiality of U is k? Prove your formula. Exercise. Develop a formula for the expasio of a triomial (x + y + z). Prove that the formula holds i a similar way to what you did i Problem. Exercise 5. Observe that the first five rows of Pascal s triagle correspod to powers of :,,,, 6. Ca you explai why? Why does t the patter cotiue after the fifth row?

5 Exercise 6. A hockeystick i Pascal s triagle is obtaied by startig o the side of the triagle (at oe of the s), movig diagoally dow for as log as you wat, ad the makig oe iety degree dowward tur ad stoppig. We call all the umbers that you traverse i the first directio the rod of the hockeystick, ad the last umber the tip. For example, if we start at ( ), we could do {,,, 5}. I this case the rod is {,, } ad the tip is 5. Show that for ay hockey stick the sum of the umbers i the rod equals the tip. 5