If a die is rolled 30 times, there are 6^{30}

Bobbie Comstock 2022-01-05 Answered
If a die is rolled 30 times, there are \(\displaystyle{6}^{{{30}}}\) different sequences possible. The following question asks how many of these sequences satisfy certain conditions.
What fraction of these sequences have exactly three 3s and three 2s?

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Expert Answer

censoratojk
Answered 2022-01-06 Author has 4897 answers
Think of a sequence as thirty empty slots.
We want to 10 slots to contain 3’s and 2’s of which we want three of the slots to be 3, the other slots to be 2. The other 20 slots can be containing any of 1, 4, 5, and 6.
As a fraction of the total of \(\displaystyle{6}^{{{30}}}\)
Firstly, select 10 slots to house the 3's and 2's - \(\displaystyle{C}{\left({30},{6}\right)}\)
Now, select 5 slots for the 3's out of 10 chosen - \(\displaystyle{\left({6},{5}\right)}\)
Select remaining 5 slots - \(\displaystyle{C}{\left({3},{3}\right)}\)
\(\displaystyle{T}{o}{t}{a}{l}={C}{\left({30},{6}\right)}\times{C}{\left({6},{5}\right)}\times{C}{\left({3},{3}\right)}\times{4}^{{{20}}}\)
\(\displaystyle{\frac{{{C}{\left({30},{6}\right)}\times{C}{\left({6},{5}\right)}\times{C}{\left({3},{3}\right)}\times{4}^{{{20}}}}}{{{6}^{{{30}}}}}}={0.01512}\) as a fraction of the total of \(\displaystyle{6}^{{{30}}}\)
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