While remodeling the house, you have 3 choices of paint color, 4 choices of carpet color, and 5 choices of furniture style. how many different groupings will you be able to make using one paint color, one carpet color, and one furniture style?

3 * 4 * 5 = 60

Justitication:

This in a direct application of the fundamental principle of counting or multiplication rule: if there are m ways of doing something and n ways of doing another thing, then there are m * n  ways of doing both actions.

In this case there are 3 choices of paint color, 4 choices of carpet color, and 5 choices of furniture style, the by the fundamental principle of counting, you have that the number of selecting the three items, using one of each is the product 3 * 4 * 5 = 6.

May be it also helps you to realize that is a combination of the elements of the set paints, with the elements of the set carpets, and the elements of the set furnitures

Paints = A = {A1, A2, A3}

Carpets = B = {B1, B2, B1}

Furnitures = C = {C1, C2, C3, C4, C5}

If you arrange the elements of each set with the elements of the other two sets, you will  see you have 3 combinations of A (color) for every comintation of the other two elements.

And you will 4 combinations of B (carpets) for every combination of C (furnitures).

That yields to 3 * 4 * 5 as stated above.

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