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# Counting problem

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An ordinary 6-sided die has a number on each face from 1 to 6 (each number appears on one face). How many ways can I paint two faces of a die blue, so that the product of the numbers on the painted faces isn't equal to 4?

Jan 4, 2022

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Explanation:

How many ways can you multiply two numbers to get 4?

Two, namely 2 x 2 and 1 x 4.

Except if you're painting two faces of a die, then you can't paint the 2 side twice, leaving only one possibility left, 1 and 4. There is only one way to paint the sides of 1 and 4.

But its not asking for the number of ways that you can paint the sides so that the product is equal to four, its asking for the opposite, reducing this question into: "How many ways are there to paint two faces of a die blue?"

$$6\choose{2}$$$$=$$$$\frac{6*5}{2*1}$$$$=15$$

Now all that needs to be done is subtract one from that to get $$14$$.

Jan 4, 2022