Find the number of ways in which twelve different baseball cards can be given syix2lns f 4ibx7+:+8:udpc- rknrb6to Emily, Harry, John and Olivia, if Emily is to receive 5 cards, Harry is to s:xl -+y6 4rpi+ cunxs8k:dr f2ib 7nbreceive 3 cards, John is to receive 3 cards and Olivia is to receive 1 card.   

  Tyler needs to decide the order in which to schedule 11 ex7lo5ddrs2n5i gy d oz:( -x5x;*kjqicams for his school. Two of these exams are Chemistry ( 1 SL and 1 HL). Find the number of different wak xr cj:;(-dyxoiznl ogi5 q27d *s55dys Tyler 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 is selected from 8 boysg853the. yx6 dawd;a 7 yqdpk0 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 female of )/tqc dojhwa(m *5+nificers. A special group of 5 officers is to be assembled for an undercover oc hw *to(+qj5 mdi/an)peration.
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 different paintbdz anq+9l ecuw: f/m.)z fz/-ings by Van Gogh, and 3 different paintings by Rembrandt. The curat:f.- n/wq/blc)zzf 9 m+ezdauor 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 hisy6xx(; wlo5rdzvm:k0 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 Chemistry subjecty;l r zo6xk(0v5wd:mxs must not be consecutive.$\approx a \times 10^{b}$ a =    b =   

  Ten students are to be arranged in a new chemistry lab. Ti2 occvv4t34nhr28 0 vz*syrmxp35bnhe chemistry lab is set out izvhn5m3y0xr2 42 r3 vb*nc8o tp4vi scn 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 o uta1 vum0 ;5-jmslx9jffering a special package on personalized keychains. Customers can choose to personalize their keychains with up to 3 different charms f591t;au u j0lmvsmx- jrom 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 a lecture seriyy18w obssg5+u, p y.2q (*osxu2dicxes. They have a row of 8 seats to themselves.2by2yx(dq8 *o os.1gs,yx +scupwui5
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 hxc57 dyz n8ei1as 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 up the nine books on the main shelf, yc7en5 x8dz1i 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 race wherec;dvcl 4b7:rf there are no tied finishes and there is a total of 10 competitors. Find the total number of possible ways in which the ten swimmers can finish if Zoe fr;cv47flbc: dinishes
1. in the position immediately after Sophia;   
2. in any position after Sophia.   

  There are 11 players on a football team wyu iz4z7s6m q7/(l k sj*j2-k bsio:gzg (g;ffho are asked to line up in one straight line for a team photo. Three of the team members named Adam, Brad and Chris refu7 sg(4bkf*zus 6g7k-gmj ls( 2fqjy:/; oizzise 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 with tws3( nc,dkezm04a v0 mlo rows and three columns as shown in the followidmlcvk0e4 0n,3s( a mzng diagram. 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 fourb7am (evtbr 6z-b5x 4 hg/ck6f 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. Th;ny , p8: subud)qu j5r(,cl 8k0fkxuxey each have five cards showing roman numerals I, V, X, L, C. Sophie lays her cards face up on the table in obnfxk8,0kusl5uu8j,u qcy:)( dr ;xp rder I, V, 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.stm : iha:(cv56dsh/-ngi j -euc4n0a They 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 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 hnvdti ae0:c6 m:-s5cj4 shg/-(ian umoney 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 .