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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 A botanist is studying a garden with $5$ distinct species of flowers. If she wishes to plant $3$ flowers in a row, with each flower being from a different species, in how many distinct sequences can she arrange the flowers?

📌 Next, for each selection of $3$ species, we arrange them in a sequence. The number of permutations of $3$ flowers is $3!$ (since each species contributes exactly one flower):

📌 Thus, the total number of distinct sequences is the product of selecting the species and arranging them:

📌 A geologist is analyzing rock samples, each with a unique mineral composition. If there are $4$ distinct types of minerals in a sample, and she selects $2$ minerals at a time for analysis, how many different pairs of minerals can she choose?

📌 The problem requires us to determine the number of ways to choose $2$ minerals from $4$ distinct types. This is a combination problem, where order does not matter, and is calculated using the combination formula:

📌 An astrophysicist is observing a cluster of $6$ stars, each with a unique spectral signature. If he wants to form a constellation by selecting $4$ stars, in how many different ways can he arrange these $4$ stars in sequence?

📌 First, we select $4$ stars from the $6$ available. The number of ways to choose $4$ stars from $6$ is given by the combination formula:

📌 After selecting $4$ stars, we arrange them in a sequence. The number of permutations of $4$ stars is:

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