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k e l l y r i z z o - AIKO, infinite ways to autonomy.

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 \binom{7}{4} = \frac{7!}{4! \times (7-4)!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35

📌 Thus, there are \(\boxed{35}\) different sets of 4 instruments that can be chosen from the 7.

📌 Question:** An agricultural systems officer in California is planning a community program and needs to select 3 types of sustainable crops from a list of 10. In how many ways can they choose 3 crops if the order does not matter?

📌 Solution:** This is another combinations problem, where we need to choose 3 crops from a list of 10. The number of combinations is given by:

📌 \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8 \times 7!}{3! \times 7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120

📌 Therefore, there are \(\boxed{120}\) ways to choose 3 crops from the list of 10.

📌 Question:** A chemist at Caltech is testing combinations of 4 different catalysts for a reaction from a set of 9 available catalysts. How many distinct sets of 4 catalysts can be selected?

📌 Solution:** This is a combinations problem where we need to choose 4 catalysts from 9. The number of ways to do this is given by:

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