LESSON 4 COMBINATIONS WARM UP: 1. 4 students are sitting in a row, and we need to select 3 of them. The first student selected will be the president of our class, the 2nd one selected will be the vice president, and the 3rd will be the secretary. In how many ways can these students be selected? 2. How would the answer change if positions are no longer assigned? Three students need selected from the group of 4 to sit on a committee. Nov 18 10:48 PM The first example on the last slide was a, the second is called a. INVESTIGATION 1: Get into groups of 6. Figure out how many handshakes are possible if each student must shake every other students hands. When you have your answer log on to www.socrative.com enter my room number and submit your answer. Nov 18 11:15 PM 1
INVESTIGATION 2: Five students, Aaron, Bobby, Chris, Dylan, and Eric take part in a cross country race to represent their school. a) Suppose the winner of the race wins $50, the runner up wins $25, and third place is $10. The table below shows all possible ways in which the three prizes could be awarded to the 5 participants in the race. "A" stands for Aaron, "B" stands for Bobby, "C" stands for Chris, "D" stands for Dylan, and "E" stands for Eric. ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ACB ADB AEB ADC AEC AED BDC BEC BED CED BAC BAD BAE CAD CAE DAE CBD CBE DBE DCE BCA BDA BEA CDA CEA DEA CDB CEB DEB DEC CAB DAB EAB DAC EAC EAD DBC EBC EBD ECD CBA DBA EBA DCA ECA EDA DCB ECB EDB EDC Is this an example of permutations or combinations? How many ways are there to award the three prizes? Nov 19 3:00 PM b) The school has been awarded three places at a running clinic. The school coach decides to select the 3 lucky students from the ones who took part in the cross country race. Use this same table to circle the different ways the three students can be chosen. ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ACB ADB AEB ADC AEC AED BDC BEC BED CED BAC BAD BAE CAD CAE DAE CBD CBE DBE DCE BCA BDA BEA CDA CEA DEA CDB CEB DEB DEC CAB DAB EAB DAC EAC EAD DBC EBC EBD ECD CBA DBA EBA DCA ECA EDA DCB ECB EDB EDC Is this an example of permutations or combinations? How many ways are there to award the three students? Nov 19 3:08 PM 2
Notice how in Investigation part b, that the number of Combinations is equal to the number of permutations divided by or, in other words, divided by factorial. Nov 19 3:12 PM COMBINATION FORMULA A combination is a selection of objects in which order isn't important. The formula for a combination is: where n has to be greater then, or equal to, r, and n>0. Nov 18 11:19 PM 3
EXAMPLES: 1. Evaluate by first using the formula, then confirm using your calculator. a) b) Nov 19 3:13 PM 2. How many different committees of 3 people can be selected from 8 people? Nov 19 3:15 PM 4
http://www.google.ca/imgres?num=10&hl=en&safe=active&tbo=d&authuser=0&biw=1280&bih= 675&tbm=isch&tbnid=zwZYVLvnPc4QeM: &imgrefurl=http://www.mohawkcollege.ca/athletics/womens basketball.html%3fpagemode% 3DPrint&docid=RqHmCKChXVmhM&imgurl=http://www.mohawkcollege.ca/Assets/Athletics/varsity/Team% 252BPhotos/Womens%252BBasketball%252BTeam%252B2011 12.jpg&w=540&h=360 &ei=ircqum6jgipriwkfsogwba&zoom=1&iact=hc&vpx=632&vpy=381&dur=765&hovh=183 &hovw=275&tx=165&ty=206&sig=109971733282591085494&page=2&tbnh=138&tbnw=200 &start=24&ndsp=30&ved=1t:429,r:33,s:0,i:174 Combinations 1a.notebook 3. In how many ways can two players be selected from a team of twelve? Nov 19 3:16 PM 4. The athletic council decides to form a sub committee of seven council members to look at how funds raised should be spent on sports activities in the school. There are a total of 15 athletic council members, 9 males, and 6 females. The sub committee must consist of exactly 3 females. a) In how many ways can the females be chosen? b) In how many ways can the males be chosen? c) In how many ways can the sub committee be chosen? d) In how many ways can the sub committee be chosen if Ms. Jensen must be on the committee? Nov 19 3:28 PM 5
http://www.google.ca/imgres?num=10&hl=en&safe=active&tbo=d&authuser=0&biw=1280&bih=675 &tbm=isch&tbnid=bfr8ajmgdxpmkm:&imgrefurl=http://www.modernfuel.org/news/427 &docid=xjfff5fc69tgym&imgurl=http://www.modernfuel.org/files/imagecache/attached_full/files/st oryimages/ist2_493857 queen of hearts playing cards background1.jpg&w=298&h=380&ei= 8bOqUNOvNu3piQKZhIDwAw&zoom=1&iact=hc&vpx=142&vpy=160&dur=1259&hovh=254 &hovw=199&tx=134&ty=272&sig=109971733282591085494&page=1&tbnh=158&tbnw=123 &start=0&ndsp=26&ved=1t:429,r:7,s:0,i:91 Combinations 1a.notebook A standard deck of 52 cards has the following characteristics: * suits * Each suit has cards. 2 4 13 * cards are black. 2 * cards are red. and ERASE TO REVEAL queens kings * Face cards are considered to be,, Nov 19 3:31 PM CARDS EXAMPLE: 5. Poker is a card game played from a deck of 52 cards. a) How many different 5 card poker hands are possible? b) In how many of these hands will there be: 1) all diamonds? 2) 4 black cards and 1 red card? 3) 3 kings and 2 queens? 4) 3 kings? 5) four aces? 6) 5 cards of the same suit? Nov 19 3:42 PM 6
6. To play in the Super 7 lottery, you must choose 7 numbers from 1 to 47. To play in the Lotto 6 49 lottery, you must choose 6 numbers from 1 to 49. To win each jackpot, the numbers chosen must match the numbers drawn by the lottery corporation. Which jackpot are you more likely to win? http://www.google.ca/imgres?num=10&hl=en&safe=active&authuser=0&biw=1280&bih=675 &tbm=isch&tbnid=kg3sqn0yjvww6m: &imgrefurl=http://www.680news.com/news/local/article/378718 two winning lotto 6 49 ticketspurchased inontario&docid=meabrfe1ahubrm&imgurl=http://media.greenradio.topscms.com/images/06/b2/0 6573b59421a82ac0be33f4b5f84.jpeg&w=425&h=425&ei=9cerUO3cDOPtiQKEsIDoDw&zoom=1 &iact=hc&vpx=174&vpy=134&dur=1964&hovh=225&hovw=225&tx=126&ty=244&sig= 109971733282591085494&page=1&tbnh=134&tbnw=132&start=0&ndsp=20&ved= 1t:429,r:0,s:0,i:68 http://www.google.ca/imgres?hl=en&safe=active&authuser=0&biw=1280&bih=675 &tbm=isch&tbnid=zkqcke9wvm2adm:&imgrefurl=http://accesswinnipeg.com/2009/03/lottosuper 7 being retired/ &docid=b2ngksbx2qksqm&imgurl=http://farm4.static.flickr.com/3584/3325640153_ 76e145db0a.jpg&w=200&h=200&ei=PMirUNjVM WjiALYm4GQAQ&zoom=1&iact=hc&vpx= 957&vpy=80&dur=2048&hovh=160&hovw=160&tx=87&ty=168&sig=109971733282591085494 &page=1&tbnh=160&tbnw=160&start=0&ndsp=20&ved=1t:429,r:3,s:0,i:74 Nov 20 11:08 AM EQUIVALENT COMBINATIONS Discovery: Solve the follwoing questions to help find equivalent combinations. a) How many ways can a committee of 6 students be selected from 10 students? b) How many ways can a committee of 4 students be selected from 10 students? c) Why are the answers in a) and b) the same? (demonstrate with students in class) Nov 20 11:20 AM 7
Prove the identity: Nov 20 11:23 AM LAST EXAMPLE: During a Pee Wee hockey tryout, all the players met on the ice after the last practice and shook hands with each other. How many players attended the tryouts if there were 300 handshakes in all? Nov 20 11:26 AM 8
Dec 13 8:12 AM 9