Hi Matt,
Pretty much everyone has posted what could be different ways of approaching this question.
I will give my 2 cents to this question.
These types of question can be solved using combinatorial approach, but the below rule would help you to approach it better.
Probability of Complex event = Product of individual events * arrangements
Here its rolling the die five times and getting six exactly once.
So according to the above rule it is, (six)(Not six) (Not six) (Not six) (Not six) * arrangement
P(getting six) = 1/6
P(not getting six) = 5/6
So, 1/6 * 5/6 * 5/6 * 5/6 * 5/6 * arrangement.
Arrangement can be done many ways, already few have posted different ways (like selections, listing..) of arranging.
Another alternative way is,
Consider this as a word,
SNNNN,
S – Six , N – Not Six
So SNNNN can be arranged in 5!/4! Ways. That is 5 different ways.
If you are not aware of the above arrangement,
Let’s say for an example, Given a word AISHA – different ways of arranging it is 5!/ 2!. Here the numerator is
arranging of 5 letters and denominator is dividing the repetitions.
So the final answer would be 5^ 5/ 6^5.
So if you look at it finding the probability is similar but arrangements can be done in multiple ways.
So you can chose best way you are comfortable with.
Hope this is clear.
