\boxed{\dfrac{5}{22}}

February 22, 2026

$\boxed{\dfrac{5}{22}}$

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Question: A historical archive contains 4 ancient scrolls, 6 manuscripts, and 2 letters. If two items are drawn at random, what is the probability that both are manuscripts?
Solution: Total items: $ 4 + 6 + 2 = 12 $. Total ways to choose 2: $ \binom{12}{2} = 66 $. Favorable ways: $ \binom{6}{2} = 15 $. Probability: $ \frac{15}{66} = \frac{5}{22} $.
How many 8-digit positive integers have all digits either 3 or 4, and have exactly three consecutive 3's?
To solve this, we can use a combinatorial approach. We want to form an 8-digit number using the digits 3 and 4, with exactly three consecutive 3's. Let's break it down:
Position the block of three consecutive 3's:

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