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Home » Total number of non-negative integer solutions (stars and bars): - SUpost

Total number of non-negative integer solutions (stars and bars): - SUpost

📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 Pink flowers only from Rr genotype.

📌 Frequency of Rr = 1/2

📌 Expected pink = 400 × 0,5 = 200

📌 ---Question: A cybersecurity consultant needs to assign 5 distinct encryption keys to 3 different servers, ensuring each server gets at least one key. How many ways can this be done?

📌 Solution: The problem requires distributing 5 distinct keys to 3 distinct servers with each server receiving at least one key. This is a classic inclusion-exclusion problem. The total number of ways to assign the keys without restriction is $3^5$. Subtract the cases where at least one server gets no keys: $\binom{3}{1} \cdot 2^5$. Add back the cases where two servers get no keys (since they were subtracted twice): $\binom{3}{2} \cdot 1^5$. Thus, the total is:

📌 3^5 - \binom{3}{1} \cdot 2^5 + \binom{3}{2} \cdot 1^5 = 243 - 96 + 3 = 150.

📌 The final answer is $\boxed{150}$.

📌 Question: A seismologist records 7 earthquakes, 3 of which are magnitude 7.5 and 4 are magnitude 6.8. How many distinct sequences can the seismologist observe?

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