
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
<u>Set 4</u> **p10.** Eve has nine letter tiles: three  $C$ ’s, three  $M$ ’s, and three  $W$ ’s. If she arranges them in a random order, what is the probability that the string “ $CMWMC$ ” appears somewhere in the arrangement?**p11.** Bethany’s Batteries sells two kinds of batteries:  $C$  batteries for  $\$ 4 $ per package, and $ D $ batteries for $ \ $7$  per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly  $\$ 2021 $, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell?**p12.** A deck of cards consists of $ 30 $ cards labeled with the integers $ 1 $ to $ 30 $, inclusive. The cards numbered $ 1 $ through $ 15 $ are purple, and the cards numbered $ 16 $ through $ 30 $ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $ 32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck.


PS. You should use hide for answers.

Present the answer in LaTex format: \boxed{Your answer}
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