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?

Guest Jul 30, 2022

#2**0 **

Wait... I made an error...

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

#1**0 **

Each side of the cube is \(\sqrt[3]{1000} = 10\).

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, \(a + b + c = \color{brown}\boxed{558}\)

BuilderBoi Jul 30, 2022

#2**0 **

Best Answer

Wait... I made an error...

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