Find the number of ways in which twelve differeq lruf.nrb2ibcc/n j*99) pu4nt baseball cards can be given to Emily, Harry, John and Olivia, if Emily is to receive 5 cards, Harry is to receive 3 cards, John is to receive 3 b/u.f cn9i29 *cu )r4jlqnrpbcards and Olivia is to receive 1 card.   

  Tyler needs to decide the order in which lv89uwm/g wn2j 7 w2utto schedule 11 exams for his school. Two of these exams are Chemistry ( 1 SL and 1 HL). Find the number of different ways Tyler can schedule the 11 exams given that the two Chemistry subjects must not be conseg9vnuu7w 8w j22/ twlmcutive.$\approx a \times 10^{b}$ a =    b =   

  A school basketball team of 5 students is selected from 8 boys f. a)l2c-nrw band 4 girls.
1. Determine how many possible teams can be chosen.   
2. Determine how many teams can be formed consisting of 3 boys and 2 girls?   
3. Determine how many teams can be formed consisting of at most 3 girls?   

  A police department has 4 male and 7 female offigll2,ln i2i -nkri1, ncers. A special group of 5 officers is to be assembled for an undenk,g lnrlni-,2l2i 1ircover operation.
1. Determine how many possible groups can be chosen.   
2. Determine how many groups can be formed consisting of 2 males and 3 females.   
3. Determine how many groups can be formed consisting of at least one male.   

  In an art museum, there are 8 difbvqwe+3; 0y1ji4 ok yah; cv4iferent paintings by Picasso, 5 different paintings by Van Gogh, and 3 different paintings by Rembrandt. The c41k 3ec qv+vbjo;0yhwia y4i;urator of the museum wants to hold an exhibition in a hall that can only display a maximum of 7 paintings at a time.
The curator wants to include at least two paintings from each artist in the exhibition.
1. Given that 7 paintings will be displayed, determine how many ways they can be selected.   
2. Find the probability that more Rembrandt paintings will be selected than Picasso paintings or Van Gogh paintings.   

  Peter needs to decide the order in which to schedule 14 exams for his scho jww5wj 5:r7ttxe;e;gol. Two of these exams are Chemistry ( 1 SL and 1 HL). Find the number of different ways Peter can schedule the 14 exams given that the two Chemistryjxw7 5; g5e wt;jrwt:e subjects must not be consecutive.$\approx a \times 10^{b}$ a =    b =   

  Ten students are to bee.dhkn2 * ;wty arranged in a new chemistry lab. The chemistry lab is set out in two rows of five desks as shown in the following diagk;*n.etydw h 2ram.



1. Find the number of ways the ten students may be arranged in the lab.

Two of the students, Hugo and Leo, were noticed to talk to each other during previous lab sessions.   
2. Find the number of ways the students may be arranged if Hugo and Leo must sit so that one is directly behind the other. For example, Dest 1 and Desk 6 .   
3. Find the number of ways the students may be arranged if Hugo and Leo must not sit next to each other in the same row.   

An arts and crafts store is offering a yy qn 8w)ezg50:qk8wlspecial package on personalized keychains. Customers can choose to personalize their keychains with up l 5808kew wgnyy q:)zqto 3 different charms from a selection of 6 different types of charms.
Determine how many ways a customer can personalize a keychain if
1. The order is important.
2. The order is not important.

  A professor and five of his students attend a talk given a*tyu d-pd t1- q7l/:rjrsi9r in a lecture series. They have a row of 8 seats to thera9/sdrj pl*r1iuqtdt-y7 :-mselves.
Find the number of ways the professor and his students can sit if
1. the professor and his students sit together.   
2. the students decide to sit at least one seat apart from their professor.   

  Julie works at a book sl/p7it ymat5v (p ie;4 srr1ye3gyg/5tore and has nine books to display on the main shelf of the store. Four of the books are non-fiction and five are fiction. Each book is different. Determine the number of possible ways Julie can line u3t (e;7 ye li mspy5tgaryr4/1/v p5igp the nine books on the main shelf, given that
1. the non-fiction books should stand together;   
2. the non-fiction books should stand together on either end;   
3. the non-fiction books should stand together and do not stand on either end.   

  Sophia and Zoe compete -mw ; ekk6s48yewvo1yin a freestyle swimming race where there are no tied finishes and there is a total of 1k6yy ;kws e 8-4meovw10 competitors. Find the total number of possible ways in which the ten swimmers can finish if Zoe finishes
1. in the position immediately after Sophia;   
2. in any position after Sophia.   

  There are 11 players on a football team who are asked to line up ixs**d dzjeam jgw 82;qfn843o 0exn9rn one straight line for a team photo. Three of the team members 80jx;wxn2a8ezsn f94* or3 jq*dgedm named Adam, Brad and Chris refuse to stand next to each other. There is no restriction on the order in which the other team members position themselves.
Find the number of different orders in which the 11 team members can be positioned for the photo.   

  There are six office cubicles arranged in a grid (y+rnzqfu; /t with two rows and three columns as shown in the following di(;+q tznf/yru agram. Aria, Bella, Charlotte, Danna, and Emma are to be stationed inside the cubicles to work on various company projects.
Find the number of ways of placing the team members in the cubicles in each of the following cases.
1. Each cubicle is large enough to contain the five team members, but Danna and Emma must not be placed in the same cubicle.   
2. Each cubicle may only contain one team member. But Aria and Bella must not be placed in cubicles which share a boundary, as they tend to get distracted by each other.   

  The barcode strings of a new product are created from four lett) /dj wic qtlpl(4-euw72fxm:ers $ \mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{D}$ and ten digits $0,1,2, \ldots, 9$ . No three of the letters may be written consecutively in a barcode string. There is no restriction on the order in which the numbers can be written. Find the number of different barcode strings that can be created.$\approx a \times 10^{b}$ a =    b =   

  Sophie and Ella play a game. Th ldhj3-gf d 62d;isjj1ey each have five cards showing roman numerals I, V, X, L, C. Sophie lays her cards face up on the table in order I, V, X, L, C as shown in the following diagra3 sjd2j1f;6dh gi- ljdm.


Ella shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Sophie's 4 card directly above. Sophie wins if no matches occur; otherwise Ella wins.
1. Show that the probability that Sophie wins the game is $\frac{11}{30}$ .

Sophie and Ella repeat their game so that they play a total of 90 times. Let the discrete random variable X represent the number of times Sophie wins.   
2. Determine:
1. the mean of X ;   
2. the variance of X .   

Jack and John have decided to play a game. They will be rolling a z/5qn(fycppq +f(n2 f die seven times. One roll n fcpznf52/p( +qqy(fof a die is considered as one round of the game. On each round, John agrees to pay Jack 4 dollars if 1 or 2 is rolled, Jack agrees to pay John 2 dollars if 3,4,5 or 6 is rolled, and who is paid wins the round. In the end, who earns money wins the game.
1. Show that the probability that Jack wins exactly two rounds is $\frac{224}{729}$ .
2. 1. Explain why the total number of outcomes for the results of the seven rounds is 128 .
2. Expand $(1+y)^{7}$ and choose a suitable value of y to prove that

$128=\binom{7}{0}+\binom{7}{1}+\binom{7}{2}+\binom{7}{3}+\binom{7}{4}+\binom{7}{5}+\binom{7}{6}+\binom{7}{7}$ .

3. Give a meaning of the equality above in the context of the seven rounds.
3. 1. Find the expected amount of money earned by each player in the game.
2. Who is expected to win the game?
3. Is this game fair? Justify your answer.
4. Jack and John have decided to play the game again.
1. Find an expression for the probability that John wins five rounds on the first game and two rounds on the second game. Give your answer in the form

$\binom{7}{r}^{2}\left[\frac{1}{3}\right]^{s}\left[\frac{2}{3}\right]^{t}$

where the values of r, s and t are to be found.
2. Use your answer to (d) (i) and seven similar expressions to write down the probability that John wins a total of seven rounds over two games as the sum of eight probabilities.
3. Hence prove that

$\binom{14}{7}=\sum_{k=0}^{7}\binom{7}{k}^{2}$

5. Now Jack and John roll a die 12 times. Let A denote the number of rounds Jack wins. The expected value of A can be written as

$\mathrm{E}[A]=\sum_{r=0}^{12} r\binom{12}{r}\left[\frac{a^{12-r}}{b^{12}}\right]$

1. Find the value of a and b .
2. Differentiate the expansion of $(1+y)^{12}$ to prove that the expected number of rolls Jack wins is 4 .