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Home » Up Close and Personal: Inside the Most Shocking Moments Ever Recorded - Appcentric

Up Close and Personal: Inside the Most Shocking Moments Ever Recorded - Appcentric

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 \binom{7}{3} = 35,\quad \binom{9}{2} = 36,\quad \binom{16}{5} = 4368

📌 \frac{35 \cdot 36}{4368} = \frac{1260}{4368}

📌 \frac{1260}{4368} = \frac{105}{364}

📌 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.

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