Probability Mania
A fair six-faced die is rolled six times.
The probability that the numbers on the die appear either in ascending or descending order is of the form
where
and
are co-prime integers.
Find the value of
Assumptions
- The order may not be strictly ascending or descending. For example,
is a valid sequence of numbers.
The answer is 19.
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1 solution
The tricky part is counting the number of valid successes. First, let's focus on the ascending sequence of 6 digits. Since they are in increasing order, we can denote them by the number of 1's, 2's, 3's, 4's, 5's, 6's, they have, subject to a + b + c + d + e + f = 6 . By the stars and bars, there are ( 6 6 + 5 ) = ( 6 1 1 ) = 4 6 2 such sequences.
There are also going to be 462 decreasing sequences, but we might have double counted. How many sequences are both increasing and decreasing? These are the constant sequences, which means that there are 6 of them!
Thus, the total number of favourable sequences is 4 6 2 + 4 6 2 − 6 = 9 1 8 . The probability is 6 6 9 1 8 = 8 6 4 1 7 . The answer that we're looking for is 2 4 8 6 4 − 1 7 = 1 9 .