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Home » A circle has a circumference of 31.4 cm. Calculate its radius and area. - Network4

A circle has a circumference of 31.4 cm. Calculate its radius and area. - Network4

📅 May 19, 2026 👤 scraface

May 19, 2026

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📌 Solution: The ichthyologist observes 7 fish over 7 days—2 clownfish (C), 3 parrotfish (P), and 2 angelfish (A). Since fish of the same species are indistinguishable, the number of distinct observation sequences is:

📌 Question: A circular park has 5 distinguishable trees planted around its perimeter and 3 indistinguishable benches to be placed among them. If rotations of the arrangement are considered identical, how many distinct arrangements are possible?

📌 Solution: First, fix one tree to account for rotational symmetry. Since the trees are distinguishable, fixing one eliminates rotational duplicates. The remaining 4 distinguishable trees can be arranged in $4! = 24$ ways around the circle.

📌 Now, we place 3 indistinguishable benches into the 5 gaps between the trees (since the trees form a circle, there are 5 gaps between adjacent trees). The number of ways to distribute 3 indistinguishable benches into 5 distinguishable gaps (with no restriction on how many per gap) is a stars and bars problem:

📌 Since the tree arrangements are fixed in relative order by rotation, and benches are placed in gaps (not among trees), and trees are already arranged with one fixed, the total number of distinct arrangements is:

📌 Solution: First compute each binomial coefficient:

📌 We are rolling four 6-sided dice and want the probability that exactly two dice show a number greater than 4 (i.e., a 5 or a 6).

📌 Probability a single die shows a number > 4:

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