$1$:$11$:$35$

Question $1$

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A couple is expecting their first child, and they are currently discussing what to name their baby. They have compiled a list of first names in the set$$

$=\{,,,\}$

and a list of middle names in the set

$=\{,,,,\}$

Assuming that the couple chooses a first name from $$ and a middle name from $,$ which of the following sets best characterizes the possible names for their child?

Question $1$Answer

a$.$

$()()$

b$.$

$$

c$.$

$$

d$.$

$$

Question $2$

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On your birthday, you must choose to either go out with friends or celebrate with family. If you go out with friends, then you plan to either go bowling, play mini$-$golf, or get ice cream. If you celebrate with family, you will either have a picnic or go out for a fancy dinner.

Which of the following is a decision algorithm that gives the total possible ways to celebrate your birthday?$$

Question $2$Answer

a$.$

Alternative$1$:$$Go out with friends.

$$Step $1$:$$Go bowling: $1$ outcome

$$Step $2$:$$Play mini$-$golf: $1$ outcome

$$Step $3$:$$Get ice cream: $1$ outcome

Alternative$2$:$$Celebrate with family.

$$Alternative$1$:$$Have a picnic: $1$ outcome

$$Alternative$2$:$$Have a fancy dinner: $1$ outcome$$

Total outcomes:$3$ total ways to possibly celebrate.

b$.$

Alternative $1$:$$Go out with friends.

$$Step $1$:$$Go bowling: $1$ outcome

$$Step $2$:$$Play mini$-$golf: $1$ outcome

$$Step $3$:$$Get ice cream: $1$ outcome

Alternative $2$:$$Celebrate with family.

$$ Step $1$:$$Have a picnic: $1$ outcome

$$Step $2$:$$Have a fancy dinner: $1$ outcome$$

Total outcomes:$2$ total ways to possibly celebrate.

c$.$

Step $1$:$$Go out with friends: $3$ outcomes

Step $2$:$$Celebrate with family: $2$ outcomes

Total outcomes:$6$ total ways to possibly celebrate.

d$.$

Alternative $1$:$$Go out with friends: $3$ outcomes

Alternative $2$:$$Celebrate with family: $2$ outcomes

Total outcomes:$5$ total ways to possibly celebrate.

Question $3$

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Consider a business with $12,000$ employees, each of whom must be assigned a unique identifier within the company's payroll system. The identifier, which is a $6$ digit code, is given through the following system:

$(1)$ The first character is the employee's first initial.

$(2)$ The second character is a letter between A$-$L$,$ where each unique letter corresponds to the employee's birth month $($A$=$January, B$=$February, etc.$)$

$(3)$ The subsequent $4$ digits can be any number between $0-9$

Which of the following statements is most accurate?

Question $3$Answer

a$.$

This system can generate up to $261210101010=1,040,000$ unique codes; however, common names and birth months could lead to problems with ensuring every employee has a unique identifier.$$

b$.$

This system can generate up to $261210101010=1,040,000$ unique codes, more than enough for every employee regardless of common names and birth months.

c$.$

This system can generate up to $26+12+10101010=10,048$ unique codes, too few to accommodate$$the employees at this company.

d$.$

This system can generate up to $26+12+10+10+10+10=78$ unique codes, far too few to accommodate the employees at this company.