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Home » MAD: \( \frac9,1007 \approx 1,285.71 \) - Moon Smoking

MAD: \( \frac9,1007 \approx 1,285.71 \) - Moon Smoking

📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 Question: How many 6-digit positive integers consist only of the digits 3 and 4 and contain at least two consecutive 3s?

📌 Then the desired count is $ 64 - B $

📌 We compute $ B $ using recursion. Let $ a_n $ be the number of $ n $-digit binary-like strings (digits 3 and 4) with no two consecutive 3s.

📌 Let $ a_n $ = number of valid strings of length $ n $ ending in 4

📌 Let $ b_n $ = number of valid strings of length $ n $ ending in 3

📌 Thus, number of 6-digit numbers with at least two consecutive 3s:

📌 Question: Three distinct prime numbers less than 40 are selected at random. What is the probability that their sum is even?

📌 Solution: First, note: the only even prime is $ 2 $. All other primes are odd. The sum of three numbers is even if:

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