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📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 So, number of favorable sequences is 840.

📌 An entomologist tags and releases 7 insects: 3 beetles, 2 flies, and 2 moths. If the insects leave the observation area one at a time in a random order, with individuals of the same species indistinguishable, what is the probability that no two moths are adjacent?

📌 First, compute total number of distinct sequences (without restriction):

📌 Now, compute number of favorable arrangements where no two moths are adjacent.

📌 Step 1: Arrange the non-moth insects.

📌 We have 3 beetles (B) and 2 flies (F), total 5 insects. Number of distinct arrangements:

📌 Step 2: Place the 2 moths (M) in non-adjacent positions.

📌 In a sequence of 5 non-moths, there are 6 possible gaps (including ends) where moths can be inserted:

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