$6.$ Attack Active wants to form a five$-$member committee to investigate expanding its specialties into cryptocurrencies. How many ways are there to form a five$-$member committee from the employee directory above? Note: Include yourself in the calculation. Show and explain all steps.

$7.$ If the committee will have a Chairman, a General Secretary, and a Research Analyst, how many ways are there to form a three$-$member committee if each position is held by a different member of the committee? Note: Include yourself in the calculation. Show and explain all steps.

$8.$ Create a graph of all employees below. Let the vertices of the graph be represented by each employee$$s name. Hint: you should have $8$ vertices. If a pair of employees share a specialty, draw an edge to connect them and label the edge with the name of their shared specialty.

Project #$2$

You are put in charge of a new client, Quantum Innovations. They require your help in several areas. Make sure to help them answer all their concerns.

Note: This material is covered in module $02$ and $03.$

$1.$ Quantum Innovations have noticed a growing number of physical penetration attempts over the last few months. The number of attempts from January to June includes $1,5,9,13,17,$ and $21$ respectively. From this pattern, they expect that each month, the number of threats will be four more than the number of attacks in the previous month.

a$.$ Based on the information above, what is a recurrence relation that describes the number of attacks each month? Let n $=$ the number of months starting with n $=1.$ Include the initial amount and recurrence formula.

an $+1=$ an $+4$

a$1=1($the initial amount in January$)$

a$0=1$

a$1=1+4=5$

b$.$ Find the equation that represents the number of attacks each month without using the previous term. Show and explain all steps.

The recurrence relation is an $+1=$ an $+4$ with the initial counting an $=1.$

The number of attempts from Jan to June includes $-1,5,9,13,17,21.$

So we add $4$ because each term is obtained by multiplying the month number by $4$ or an $=4$n$.$

a$2=5+4=9$

a$3=9+4=13$

a$4=13+4=17$

a$5=17+4=21$

a$6=21+4=25$

a$7=25+4=29$

a$8=29+4=33$

a$9=33+4=37$

a$10=37+4=41$

The explicit formula for the number of attacks each month is an $=4$n$.$

c$.$ How many attacks are expected in October? Use the equation you created in part $($b$)$ for your calculations. Show and explain all steps.

a$2=5+4=9$

a$3=9+4=13$

a$4=13+4=17$

a$5=17+4=21$

a$6=21+4=25$

a$7=25+4=29$

a$8=29+4=33$

a$9=33+4=37$

a$10=37+4=41$ attacks in October

OR

Recurrence Relation:

an $=$ Cn $+$ d

an $=4$n $+$ d

$-4=-4($Minus $4$ on each side$)$

an $=1+4$n

a$10=1+410$

an $=41$

$2.$ They require software specialty support $50\%$ of the time, the server configuration specialty $18\%$ of the time, and both software and server configuration $12\%$ of the time. What percent of the time will they require either the software or the server configuration specialties? Show and explain all steps.

A be the event of requiring software specialty.

B be the event of requiring server configuration specialty.

The percentage of the time they will require either the software or the server configuration specialties can be calculated using the formula:

P$($A$\backslash$cup B$)=$P$($A$)+$P$($B$)$P$($A$\backslash$cap B$)$P$($A$\backslash$cup B$)=$P$($A$)+$P$($B$)$P$($A$\backslash$cap B$)$

P$($A$)=50\%$P$($A$)=50\%($probability of requiring software specialty$)$

P$($B$)=18\%$P$($B$)=18\%($probability of requiring server configuration specialty$)$

P$($A$\backslash$cap B$)=12\%$P$($A$\backslash$cap B$)=12\%($probability of requiring both software and server configuration$)$

Substitute the values into the formula:

P$($A$\backslash$cup B$)=50\%+18\%12\%$P$($A$\backslash$cup B$)=50\%+18\%12\%$

P$($A$\backslash$cup B$)=56\%$P$($A$\backslash$cup B$)=56\%$

Therefore, they will require either the software or the server configuration specialties $56\%$ of the time.

$3.$ Active Attack needs to decide if they will take a job from Quantum Innovations. The job requires $$15,000$ of research from Active Attack before they can start the task. The company expects that it has a $40\%$ chance of completing the task without failure. If Active Attack will earn $$175,000$ for completing the task and nothing for failing to complete the task, what is the expected value of the problem? Show and explain all steps.

The expected value of the problem can be calculated by considering the probabilities of different outcomes and their corresponding values.

Let's define the random variable X$.$ X as the profit $($or loss$)$ from taking the job. The possible outcomes are:

Completing the task successfully:

X$=175,00015,000=160,000$

Failing to complete the task:

X$=0$

The probabilities associated with each outcome are:

Probability of completing the task successfully:

P$($Success$)=40\%=0\mathrm{.}4$

Probability of failing to complete the task:

P$($Failure$)=1$P$($Success$)=10\mathrm{.}4=0\mathrm{.}6$

The expected value $($E$($X$))$ is calculated as the sum of the products of each outcome and its probability:

E$($X$)=$P$($Success$)$X$($Success$)$