Problem: Your friend Vega has created a mobile game app called Permutation Planet,

where users can collect cards featuring cartoon characters and use them to play a game.

Part $1$: Every time someone makes an account on Permutation Planet for the first time,

they are given a free loot box with $5$ collectible cards. There are $60$ common cards, $25$

uncommon cards, $12$ rare cards, and $3$ ultra$-$rare cards in the set that all can be included.

Every loot box has $5$ unique cards $($with no repeats$)$ and the cards are always presented in

alphabetical order when the box is opened no matter which cards are included. How many

possible loot boxes are there?

Part $2$: A jackpot box is a special loot box that contains all $3$ ultra$-$rare cards. How many

possible jackpot boxes are there? What percentage of all the possible loot boxes are the

special jackpot boxes? $($Round to the nearest hundredth of a percent$)$

Part $3$: Instead of allowing each user to pick their own password, your friend Vega thinks

assigning a randomly generated password to each user will be safer. When a user signs up$,$

they are given a random password that is exactly $6$ characters long. The passwords can only

contain lowercase letters from the English alphabet, capital letters from the English alpha$-$

bet, or digits $0$ through $9.$ The capitalization of the letters and order of the characters matter.

Vega can$$t decide whether or not to allow repeated characters in the passwords or not. How

many possible unique passwords are there if repeated symbols ARE allowed? How many

possible unique passwords are there if repeated symbols ARE NOT allowed?

If Vega wants to choose the option with the MOST unique passwords, do you recommend

they allow repeated symbols or not allow repeated symbols?

$1$

Part $4$: Months later, your friend Vega calls you in the middle of the night with a big prob$-$

lem. They forgot their password for their Permutation Planet account! Vega knows that

their password starts with $3$ repeated capital letters $($though they can$$t remember which

one.$)$ Vega also remembers that their password ends with $3$ numbers that are all different

from each other. If these are your only clues, how many potential passwords could Vega have?

If you and Vega work together, you can check $1$ password every $3$ seconds. What is the

longest amount of time it could take for the two of you to find Vega$$s password? $($You$$ll

want to convert your answer to hours.$)$

Part $5$: Vega has even worse news, their enemy Barnaby knows that Vega has lost their

password. Barnaby knows that Vega$$s password starts with $3$ repeated capital letters, but

luckily, they don$$t know anything about the last $3$ characters. How many potential pass$-$

words are there that start with $3$ repeated capital letters?

Barnaby has access to a hacking program that can check $150$ passwords a second. What is

the longest amount of time it could take Barnaby$$s hacking program to find Vega$$s pass$-$

word? $($You$$ll want to convert your answer to hours.$)$

Using what you found in Part $4,$ who do you think is most likely to find Vega$$s password

first? Why?