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Home » Partial Fraction Decomposition? Try Our Smart Calculator for Lightning-Fast Berefs! - Dachbleche24

Partial Fraction Decomposition? Try Our Smart Calculator for Lightning-Fast Berefs! - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 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} $.

📌 \boxed{\dfrac{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:

📌 The block "333" can start at positions 1 through 6 in the 8-digit number. This gives us 6 possible positions for the block.

📌 Fill the remaining 5 positions:

📌 After placing "333", we have 5 positions left. Each of these can be either a 3 or a 4, except that no additional block of three consecutive 3's is allowed. We need to ensure that no new block of three consecutive 3's is formed.

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