Find the number of ways in which twelve different baseball cards can be,g jgk0 1wkq:p given to Emil :qkk1gg, wp0jy, 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 ord bn36gg2nm jfm.95tflx2,8 elyr ul*y er in which to 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 consecuti8 mr,2jnl6bf 3y259 gune.* fgymxtllve.$\approx a \times 10^{b}$ a =    b =   

  A school basketball team of 5 students is selected frw(ftr la:4(s(aw b (zkom 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 has 4 male and 7 femalkut-j*gm j-xkl4x js0ld8 (,ze officers. A special group of 5 officers is to be assembled for an undercover operationk8uj-xljzd 4j st0-*,(gm kxl .
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, therf6wx w5v*ojfe*xf8.v e are 8 different paintings by Picasso, 5 different paintings by Van Gogh, and 3 different paio*. 6 wv8fxxvfe wj5f*ntings by Rembrandt. The curator 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 which9 d3tizdm4h5bsf w 4+c to schedule 14 exams for his school. 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 Chedbm49itdsz3 + 5hc4fwmistry subjects must not be consecutive.$\approx a \times 10^{b}$ a =    b =   

  Ten students are to be arrad g)3wwhdn7 t3nged in a new chemistry lab. The chemistry lab is set out in two rows of five desks as shn7h3wgd)3 tdw own 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 special package on personalized keychaib q1k7 w6cq*jbns. Customers can choose to perscbbqj6wk*q1 7onalize their keychains with 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 talk given in rto3l3x u(jev k7nsnh(72 x;pqxe. r ;a lecture series. They have a row of 8 seats to thnxp(3ek7;(v7e x2x ho;.l urrsqjtn3emselves.
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 onckvsvt,2 pti u48.o h+ 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 oht. +2, vkt cv8isp4uup 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 swimming rac( m0eftoc*c3)g*m oqte where there are no tied finishes c3) (*om ceqt0g*tfm oand there 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 m/zj rd5m*s:y *zxp8rvv8u(r on a football 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 toud8 szzr :vyr/x8j *rmv*pm5 ( 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 with two rows an;5fyb 2o jzkiqb;+d27pp n(u rz avp6:d three columns as shown in the following diagram. Aria, Bella, Charlotte, Danna, and Emma are to be stationed inside the cubicles to work on various company projectsk7z :nb u;pdf (r6 zoypj2qi+2a5b; pv.
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 letterh n-d7p1vjz o,y8y4j ls $ \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 cards shksil4bj: r. icn,+jw*yyn n0 7owing roman numerals I, V, X, L, C. Sophie lays her cards face up on the table in order I, Vn :biji k7wy +sl0*cr.nnjy,4 , X, L, C as shown in the following 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. Theytru*88v6c hdo will be rolling a die seven times. One roll of a die is considered as one round of the game. On each round, John agrees to pay Jack 4 dollars *h8ou8tdrvc6 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 .