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Home » Brownie Mix Cookies That Taste Like Pure Chocolate Heaven – Try Them Now! - Veritas Home Health

Brownie Mix Cookies That Taste Like Pure Chocolate Heaven – Try Them Now! - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Thus, the probability is $\boxed{\frac{105}{364}}$.

📌 Question: How many 8-digit positive integers composed only of the digits 3 and 4 contain at least one instance of two consecutive 3s?

📌 Solution: First, compute the total number of 8-digit numbers using only digits 3 and 4:

📌 Now count the number of such numbers that do **not** contain two consecutive 3s. Let $a_n$ be the number of valid sequences of length $n$ with no two consecutive 3s, where each digit is 3 or 4.

📌 We define a recurrence:

📌 If the first digit is 4, the remaining $n-1$ digits form a valid sequence: $a_{n-1}$

📌 If the first digit is 3, the next must be 4, and the rest $n-2$ digits form a valid sequence: $a_{n-2}$

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