1. Explain the difference between a permutation and a combination. Illustrate these 2 concepts with an example. f3 2. A teacher is decorating their classroom by placing posters in a row on one wall. The teacher has 12 posters, of which 4 are colored and 8 are black 3: white. a} If the teacher decides to only use 6 posters, determine how many arrangements can be made? Describe 2 different ways to get an answer to this problem. t2 b} If the teacher uses all 12 posters, but wants to have a black (5:: white poster at each end, how many arrangements can be made? Justify your answer. I\" l c} Ifthe teacher uses all 12 posters, but wants all the colored posters to be side-by- side, how many arrangements can be made? Justify your answer. I] 3. A bookstore is putting together a gift basket of 11 books for a silent auction prize. They would like to include 7 fiction books and 4 non-fiction books from the best-seller list. Right now, there are 15 fiction books and 10 non-fiction books on the best-seller list. Tony thinks that 2567 X 25C4 is the correct way to find the number of gift baskets. Jenny thinks that 157 + 10C4 is the correct way to find the number of gift baskets. a) Both Tony and Jenny are incorrect. Explain each of their mistakes. 12 b) Correctly determine the number of gift baskets that can be made. /14. A group of 5 grade 11 students and 16 grade 12 students apply to go to a student conference in Vancouver, but there is only enough funding to send 6 students. a) Determine how many ways the 6 students can be selected if equal numbers of grade 11's and grade 12's are chosen? /1 b) Determine the number of groups that can be created if at least three of 14 the students must be in grade 12? Solve this problem using 2 different methods. c) Determine the probability that the group of 6 students sent to Vancouver would consist of no more than 2 grade 12 students. 12 5. Lotto 6/49 is a Canadian lottery game where you select a certain combination of 6 numbers from 1-49 to earn a chance at winning a cash prize. There are no repetitions of numbers here. Additionally, you must select a bonus number. There are certain cash prizes you can win that depend on how many correct numbers you select on your ticket. Here is an example of the prize breakdown of the main draw from Saturday, January 16, 2021: LOTTO 6/49 Prize Breakdown Saturday, January 16, 2021 Jackpot: $5,000,000.00 MAIN DRAW onus 9 10 21 28 35 43 34 Prize Category Winners Prize 6 of 6 CARRIED OVER 5 of 6 + Bonus NOT WON 5 of 6 $2,157.50 4 of 6 4.105 579.90 3 of 6 17.169 $10.00 2 of 6 + Bonus 18 353 $5.00 2 of 6 530,406 FREE PLAY GUARANTEED PRIZE DRAW Prize Breakdown Exact Match Only Winning Number Winners Prize 18396220-0 1 - Ontario $1,000,000 00 EXTRA Prize Breakdown Saturday, January 16, 2021As you can see, there are seven prize categories: 1. 6 of6 {you selected all 6 numbers correctly} 5 of 6 + bonus {you selected 5 of 6 numbers correctly and the bonus} 5 of 6 {you selected 5 of 6 numbers correctly} 4 of 6 {you selected 4 of 6 numbers correctly} 3 of 6 {you selected 3 of 6 numbers correctly} 2 of 6 + bonus {you selected 2 of 6 numbers correctly and the bonus} 2 of 6 {you selected 2 of 6 numbers correctly} HP'WPP'E'J a] Is this lottery an example of a permutation or a combination? Explain. fl b] If you purchased one Lotto 6649 ticket, determine the probability that you would win the main prize \"6 of 6\"? Leave your answer as a fraction. II c] If you purchased ten Lotto tiff-19 tickets, determine the probability that you would win the main prize \"6 of 6\"? Leave your answer as a fraction. 3' l d] If you purchased one Lotto 6'49 ticket determine the probability that you I 1 would win the prize \"5 of 6\"? (Hint: To match 5 winning numbers you must select 5 of 6 winning numbers AND you must select 1 number from the other 43 non-winning numbers.)