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hyatt regency sydney sydney nsw - Imagemakers

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 We want to count the number of binary strings of length 8 with exactly three 1s, no two of which are adjacent.

📌 To ensure no two 1s are adjacent, we model this as placing 3 ones into 8 positions such that there is at least one 0 between any two 1s.

📌 Step 1: Place the 5 zeros first. This creates 6 possible gaps where 1s can be placed: before the first zero, between zeros, and after the last zero.

📌 We need to choose 3 of these 6 gaps to place exactly one 1 in each (to avoid adjacency).

📌 Each such placement gives a unique binary string with 3 non-adjacent 1s.

📌 Thus, the number of valid strings is:

📌 Total labor = 45 hours + 50 hours = <<45+50=95>>95 hours

📌 A science experiment requires mixing three solutions: Solution X, Solution Y, and Solution Z. The amount of Solution X is twice that of Solution Y, and the amount of Solution Z is three times that of Solution Y. If Solution Y is used in a 4-liter quantity, how many liters of solution are used in total?

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