Find the number of ways in which twelve different basebagyh 33 xw; ;yzjfo5,w5wnbr,sll cards can be given to Emily3,sz w,ohy w n;b55;rwy j3xfg, Harry, John and Olivia, if Emily is to receive 5 cards, Harry is to receive 3 cards, John is to receive 3 cards and Olivia is to receive 1 card.   

  Tyler needs to decide the order in which totyh28) jnn,o) )v-jde; kqrvj rb3:t t schedule 11 exams for his school. Two of these exams are Chemistry ( 1 SL and 1 HL). Find the number of different ways Ty)3t;krbj )-vntt) vrq,jdnje :yh 2o 8ler can schedule the 11 exams given that the two Chemistry subjects must not be consecutive.$\approx a \times 10^{b}$ a =    b =   

  A school basketball team of 5 students i1 fjm ou3t30uis selected from 8 boys and 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 ha -q8pl6r wt;d2n3fpahyylu us;.3,ams 4 male and 7 female officers. A special group of 5 officers is to be assembled for an undercover operation.-hp8q .ldl prs,fyt3a n;6u23ya umw;
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 different paintings by Picasso, 5 differxur;3 x lnutw1 l 62+:je7iyocent paintings by Van Gogh, and 3 different paintings by Rembrandt. The curator of the museum wants to hold an exhibition in a hall that can only3xe w+l2:tl ;xn16 jyo7urciu 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 wh 3hx*,kory *e p9u(ea0hvx m-zich to schedule 14 exams for his school. Two of these exams are Chemist,ea *hyxmu 0z xp(h9* over3k-ry ( 1 SL and 1 HL). Find the number of different ways Peter can schedule the 14 exams given that the two Chemistry subjects must not be consecutive.$\approx a \times 10^{b}$ a =    b =   

  Ten students are to be arranged in a new chemistry lab. The chemistrdgftgq:8zx lq0 j,n)4 ngr +0ty lab is set out in 80nldq0gzn , 4gxrtqf tj+ )g:two rows of five desks as shown in the following diagram.



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 sou z1 .so)363rgr4cz0n erorqpecial package on personalized keychains. Customers can choose to personalize their keychains witz 30orsg4z.6euro3 1)qon r crh up to 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 talkd.v/57nx pr jb given in a lecture series. They have a row of 8 seat.rn/5 7xjbvp ds to themselves.
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 store and has nine books to display o4tn6p4 slotx 6n the main shelf of the store. Four of the books are non-fis46nx4ptt6l oction and five are fiction. Each book is different. Determine the number of possible ways Julie can line up 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 in a freestyle swij92a clz5o+ ce ,o4knmmming race where there are no tied finishes and thl+4zac2 n 5e,jkmo9oc ere is a total of 10 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 fooasjgn ;lz.yt9sr36c2 s/+k(x-gy.j0ui khmbtball team who are asked to line up in one straight line for a team photo. Three of the team members named Adam, Brad and Chris refuse to stand next to each other. The.jm syk+(.ljy -zc/i h6gr; k nbs9xtaus02g 3re 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 cubi/ w u9f-o1t 5mdr29eky ha6tcocles arranged in a grid with two rows and three columns as shown in the following diagram. Aria, Bella, Charlotte, Danna, and Emma are to be stationedc - r/21kah96u 9oedfot5mywt 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 fourm6lwt3)uep1glu9qv:luv1h t (e no)gy- :5sw letters $ \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. They each have five car/ osfqpt9 53kwp9cji8- jfnm5 0vo4ez ds 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 fq4j5fen5svk0moop89 93 j tip/c- zwf ollowing diagram.


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 wkdm04lyg mw:+ill be rolling a die seven times. One roll of a dimmkyg +dlw40: e 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 .