What's the formula for number of configurations to five shapes with colors

Question

I have triangle, circle, square, diamond, and trapezoid, each with 36 colors. How do I find the possible number of unique configurations of these colored shapes.

Example: One configuration is red triangle, red circle, red square, blue diamond, blue trapezoid. Another configuration is red triangle, red circle, red square, GREEN diamond, GREEN trapezoid, etc, etc.

Would this formula work? (36!)^4

However, (36!)^4 is over a billion! Can 5 shapes with 36 colors each create over a billion configurations?

Edit: Order of the shape matters and you cannot repeat the shapes, but you can repeat colors. What matters is a combination does not appear twice.

Answer 1

The end result, whatever the interpretation of the problem is, the solution relies on the Rule of Product(Multiplication Principle).

In set theory it is written as $|A\times B|=|A|\times |B|$

(the $\times$ on the left hand of the equation is the cartesian product of sets, whereas on the right it is the usual multiplication of real numbers)

For enumerative combinatorics (e.g. counting how many possibilities there are for something) it can be paraphrased in the following way:

When attempting to count how many outcomes there are for a given scenario, if you can uniquely describe each possible outcome via a sequence of steps such that

• Every outcome is described by exactly one sequence of choices
• The number of options for each step in the process is independent of the specific choices picked in earlier steps

the total number of outcomes is then the product of the number of options available in each step.

In your specific example, we break it down via multiplication principle using the steps:

• Pick the color of the triangle (36 choices)
• Pick the color of the circle (36 choices)
• Pick the color of the square (36 choices)
• Pick the color of the diamond (36 choices)
• Pick the color of the trapezoid (36 choices)

The total is then the product of the number of choices, $36\cdot 36\cdot 36\cdot 36\cdot 36=36^5$

NOTE: The phrasing I used only says that the number of choices at each step isn't allowed to change based on earlier selections, however the specific choices available themselves are able to change.

For example, trying to arrange the five shapes in a line (no color for this example)

• Pick the location of the triangle (5 choices) (specifically the first spot, second spot, third spot,...)

• Pick the location of the circle (4 choices) (specifically if the triangle was in the first spot the circle could go in positions 2,3,4,5. If the triangle was in the second spot the circle could go in positions 1,3,4,5. If the triangle was in the third spot the circle could go in positions 1,2,4,5. If the triangle was......)

• Pick the location of the square (3 choices) (if the triangle and circle were in the first spot and second spot respectively the square could go in positions 3,4,5. if the triangle and circle were......)

• $\vdots$

Multiplying we get $5\cdot 4\cdot 3\cdot 2\cdot 1=5!=120$ ways we can arrange the five shapes. Note here in this example, at each step, the number of options was always the same regardless which specific choices were made in previous steps. Regardless which two spaces the triangle and circle occupied there were always three options available for where the square could be placed.

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For a final example, we ask the question for how many arrangements of the five shapes with 36 available colors for each shape where colors can be repeated but shapes could not where shapes could appear in any order and order matters there are.

Breaking apart via multiplication principle yet again, we get the sequence of steps:

• Pick how the shapes are each colored ($36^5$ options. See first example)
• Pick how the shapes are arranged in the line ($5!$ options. See second example)

We get here $5!\cdot 36^5$ possible outcomes.