Bobbie Comstock

2022-01-05

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

censoratojk

Expert

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 ${6}^{30}$
Firstly, select 10 slots to house the 3's and 2's - $C\left(30,6\right)$
Now, select 5 slots for the 3's out of 10 chosen - $\left(6,5\right)$
Select remaining 5 slots - $C\left(3,3\right)$
$Total=C\left(30,6\right)×C\left(6,5\right)×C\left(3,3\right)×{4}^{20}$
$\frac{C\left(30,6\right)×C\left(6,5\right)×C\left(3,3\right)×{4}^{20}}{{6}^{30}}=0.01512$ as a fraction of the total of ${6}^{30}$

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