(a)

The given information:

-The number of suspects is 6

-The number if weapons is 6

-The number of rooms is 9

One of each is randomly chosen. The object of the is to guess the chosen three.

Let us find the total number of possible solutions.

The total number of solutions = The number of suspects * The number of weapons* The number of rooms=6*6*9=324

(b)

When the selection is made, each of the players is randomly given three of the remaining cards

-Let S represents the number of suspects in the set of three cards

-Let W represents the number of weapons in the set of three cards

-Let R represents the number of rooms in the set of three cards

-Let x represents the number of solutions that are possible after a player observes given three cards.

X=(6-S)*(6-W)*(9-R)

(c)

The value of \(\displaystyle{S},{W},{R}\Rightarrow{\left\lbrace{0},{1},{2},{3}\right\rbrace}\)

And we have, S+W+R=3

{3,0,0},{0,3,0},{0,0,3} {1,1,1},{2,1,0},{2,0,1},{1,2,0},{0,2,1},{1,0,2},{0,1,2}, these are combinations of 3 cards.

So, there are 10 possible combinations of 3 cards.

E[X]=1/10sum_Ssum_Wsum_R(6-S)*(6-W)*(9-R)=1/10sum_S(6-S)sum_W(6-W)sum_R(9-R)=1/10[6sum_(W+R=3) (6-W)(9-R)+5sum_(W+R=2) (6-W)(9-R)+4sum_(W+R=1) (6-W)(9-R)+3sum_(W+R=0) (6-W)(9-R)]=190.4

The given information:

-The number of suspects is 6

-The number if weapons is 6

-The number of rooms is 9

One of each is randomly chosen. The object of the is to guess the chosen three.

Let us find the total number of possible solutions.

The total number of solutions = The number of suspects * The number of weapons* The number of rooms=6*6*9=324

(b)

When the selection is made, each of the players is randomly given three of the remaining cards

-Let S represents the number of suspects in the set of three cards

-Let W represents the number of weapons in the set of three cards

-Let R represents the number of rooms in the set of three cards

-Let x represents the number of solutions that are possible after a player observes given three cards.

X=(6-S)*(6-W)*(9-R)

(c)

The value of \(\displaystyle{S},{W},{R}\Rightarrow{\left\lbrace{0},{1},{2},{3}\right\rbrace}\)

And we have, S+W+R=3

{3,0,0},{0,3,0},{0,0,3} {1,1,1},{2,1,0},{2,0,1},{1,2,0},{0,2,1},{1,0,2},{0,1,2}, these are combinations of 3 cards.

So, there are 10 possible combinations of 3 cards.

E[X]=1/10sum_Ssum_Wsum_R(6-S)*(6-W)*(9-R)=1/10sum_S(6-S)sum_W(6-W)sum_R(9-R)=1/10[6sum_(W+R=3) (6-W)(9-R)+5sum_(W+R=2) (6-W)(9-R)+4sum_(W+R=1) (6-W)(9-R)+3sum_(W+R=0) (6-W)(9-R)]=190.4