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# Jaylin has a wooden cube which is painted blue on the outside. She cuts the cube into $1000$ identical cubes, some of which have some sides

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Jaylin has a wooden cube which is painted blue on the outside. She cuts the cube into  1000 identical cubes, some of which have some sides painted blue, then rolls the resulting cubes like dice. The probability that no blue faces land up after Jaylin rolls the  1000 cubes can be expressed as  2^a * 3^b * 5^c where a,b and  c are integers. What is the value of a+b+c?

Jul 30, 2022

#2
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The probability is $$(5^{394} \times 2^{96}) \times 2^{-392} \times 3^{-470} = 5^{384} \times 2^{-296} \times 3^{-470} = \color{brown}\boxed{-382}$$

Jul 31, 2022

#1
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Each side of the cube is $$\sqrt{1000} = 10$$.

There are 512 cubes inside that have no faces painted, so the probability is $$1^{512} = 1$$

There are 8 corners, each with a probability of $${3 \over 6} = {1 \over 2} = 2^{-1}$$, so the probability is $${1 \over 2}^8 = {1 \over 2^8}$$

There are $$8 \times 12 = 96$$ edge pieces (there are 12 edges in a cube, each edge has 8), and each cube has a probability of $${4 \over 6} = {2 \over 3}$$, so the probability is $${2 \over 3}^{96} = {2^{96} \over 3^{96}}$$

There are $$8 \times 8 \times 6 = 384$$ center cubes, each with a probability of $${5 \over 6}$$, so the probability is $$\large{{5 \over 6}^{384} = {5^{384} \over 6^{384}} = {{5^{384}} \over 2^{384} \times 3^{384}}}$$

So, the probability is $$\large{{1 \over 2^8} \times {2^{96} \over 3^{96}} \times {5^{384} \over 2^{384} \times 3^{384}} = {5^{384} \times 2^{96} \over 2^{392} \times 3^{470}} = {(5^{384} \times 2^{96})} \times 2^{-392} \times 3^{470} = 5^{384} \times 2^{-296} \times 3^{470}}$$

Thus, $$a + b + c = \color{brown}\boxed{558}$$

Jul 30, 2022
#2
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The probability is $$(5^{394} \times 2^{96}) \times 2^{-392} \times 3^{-470} = 5^{384} \times 2^{-296} \times 3^{-470} = \color{brown}\boxed{-382}$$

BuilderBoi  Jul 31, 2022
#3
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sorry, it is incorrect, im not sure why tho

Guest Jul 31, 2022
#5
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I'm with  BuilderBoi apart from one small slip.

3^96 * 3^384 = 3^480, not 3^470.

That gets  the result  a + b + c = -392.

Guest Aug 2, 2022
#4
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Since probabilities must be less than 1, I've interpreted the question as follows: Aug 2, 2022