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So, favorable outcomes: $6 imes 56 = 336$.
Question: An elementary student is building a model with 3 red blocks and 3 blue blocks, arranging them in a line. How many distinct color patterns can be formed if no two adjacent blocks may be the same color?
Solution: Since no two adjacent blocks can be the same color and we have equal numbers of red and blue blocks, the only possible alternating patterns are RBRBRB and BRBRBR. These are the only two sequences that satisfy the condition and use 3 red and 3 blue blocks. No other arrangement avoids adjacent duplicates. For example, starting with B leads to BRBRBR, and starting with R leads to RBRBRB. No other starting point avoids conflict. Thus, there are exactly 2 valid arrangements.
Solution: The area $ A $ of a right triangle is given by $ A =
Question: A meteorologist models a storm system as a circular region with radius $ r $, and the volume of rainfall is modeled as a hemisphere with the same radius. If the volume of this hemisphere is equal to the volume of a cone with height $ h = 3r $, what is the ratio of $ r $ to $ h $?
Solution: The volume of a hemisphere of radius $ r $ is:
The volume of a cone with radius $ r $ and height $ h = 3r $ is:
This is a contradiction unless we reinterpret the question: the volumes areEqual, so we solve:
Question: A mammalogist observes a group of monkeys forming a triangle while foraging in a forest clearing. The triangle has side lengths of 5 meters, 6 meters, and 7 meters. Find the radius of the circle that can be inscribed in this triangle.
Solution: The semiperimeter $ s $ of the triangle is: