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Home » \[ 2 = \fraca + b - 102 \] - High Altitude Science

\[ 2 = \fraca + b - 102 \] - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Frage: Wie viele 8-stellige positive ganze Zahlen bestehen nur aus den Ziffern 2 und 3 und enthalten mindestens drei aufeinanderfolgende 2er?

📌 Lösung: Wir suchen die Anzahl von 8-stelligen Zahlen, bei denen jede Ziffer entweder 2 oder 3 ist, und es mindestens drei aufeinanderfolgende 2er gibt (also „222 als Teilstring).

📌 Sei $S$ der Gesamtanzahl solcher Zahlen: $2^8 = 256$, da jede Stelle 2 Möglichkeiten hat.

📌 Sei $A$ die Anzahl der 8-stelligen Binärstrings (with digits 2 and 3) that **do not** contain 222 as a substring.

📌 We compute $A$, then subtract from 256:

📌 \text{Answer} = 256 - A

📌 Wir definieren eine Rekursion für die Anzahl der Strings der Länge $n$ über $\{2,3\}$ ohne „222.

📌 Let $a_n$ = number of such strings of length $n$ with no three consecutive 2s.

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