Problem

A \(12\text{-}by-8\text{-}by-10\text{-}\text{inch}\) cabinet receives \(3\text{-}by-2\text{-}by-5\text{-}\text{inch}\) bins in the corresponding orientation shown. Calculate the whole-bin count and remainder along each axis, state whether the bins fit, and find the total count.

An aligned cabinet and sample bin are dimensioned 12 by 8 by 10 inches and 3 by 2 by 5 inches in matching orientation; counts and fit decision are withheld.

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The axis counts are floor(12/3)=4, floor(8/2)=4, and floor(10/5)=2, with remainders 0, 0, and 0. The bins fit in the stated orientation, and 4(4)(2)=32 bins fit.

Using only the first two axes gives floor(12/3)floor(8/2)=4(4)=16 bins, with the 10-inch height and 5-inch bin dimension omitted.

The axis counts are 4, 4, and 2 with remainders 0, 0, and 0, but adding the counts gives 4+4+2=10 bins.

Reordering each bin as 5 by 2 by 3 gives counts floor(12/5)=2, floor(8/2)=4, and floor(10/3)=3, for 2(4)(3)=24 bins in a different orientation.

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A \(12\text{-}by-8\text{-}by-10\text{-}\text{inch}\) cabinet receives \(3\text{-}by-2\text{-}by-5\text{-}\text{inch}\) bins in the corresponding orientation shown. Calculate the whole-bin count and remainder along each axis, state whether the bins fit, and find the total count.

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