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📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 Question: A GIS analyst is analyzing satellite data from 9 time points and wants to select 4 non-consecutive time intervals for detailed heat zone analysis. How many ways can she choose 4 non-consecutive time points from the 9?

📌 Lets define the positions as 1 through 9. We want to choose 4 positions such that no two are adjacent.

📌 This is equivalent to choosing 4 positions from 9 such that there is at least one gap between any two selected positions.

📌 To model this, imagine placing 4 selected positions (denoted S) and 5 unselected positions (denoted U), such that no two Ss are adjacent.

📌 To ensure non-consecutiveness, we must place at least one U between each pair of Ss. That is, we place 3 Us to separate the 4 Ss.

📌 Remaining 2 Us are free to be placed anywhere (before, between, or after the Ss)

📌 This is equivalent to the number of integer solutions to:

📌 Question: A science teacher uses colored balls to demonstrate probability. She has 5 red, 4 blue, and 3 green balls, all indistinguishable within colors. In how many distinct ways can she arrange all the balls in a line so that no two green balls are adjacent?

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