Siuan's grandmother recently signed up for a twitter account. She only follows five people - her five grandchildren. Each day, each of her grandchildren make two tweets and at the end of the day Siuan's grandmother gets an e-mail that lists all ten tweets in chronological order. If the times of each grandchild's tweets are random, the probability that no consecutive pair of tweets in the e-mail are by the same person can be expressed as a/b where a and b are coprime positive integers. What is the value of a + b?
fail.
First, we will find the probability of at least one grandchild making consecutive tweets by using the inclusion-exclusion principle, and then subtract the result from 1.
From the inclusion-exclusion principle,
P(at least one grandchild makes consecutive tweets)
= (5 choose 1)P(a specified grandchild makes consecutive tweets) - (5 choose 2)P(two specified grandchildren makes consecutive tweets) + (5 choose 3)P(three specified grandchildren makes consecutive tweets) - (5 choose 4)P(four specified grandchildren makes consecutive tweets) + (5 choose 5)P(all five grandchildren makes consecutive tweets).
For 1 <= k <= 5, we need to find the probability that k specified grandchildren (and possibly others) make consecutive tweets.
There are 10! permutations of all 10 tweets.
Think of arranging k "blocks" of two tweets each, and (10 - 2k) single tweets. There are (10 - k)! ways of arranging these (10 - k) items, and 2 ways of arranging the two tweets within each of the k blocks. So (10 - k)!(2^k) of the 10! permutations result in k specified grandchildren (and possibly others) making consecutive tweets.
So the probability that k specified grandchildren (and possibly others) make consecutive tweets is (2^k)/[10*9*...*(10 - k + 1)].
Therefore, we have
P(at least one grandchild makes consecutive tweets)
= 5(2/10) - 10(4/(10*9)) + 10(8/(10*9*8)) - 5(16/(10*9*8*7)) + 1(32/(10*9*8*7*6))
= 1 - 4/9 + 1/9 - 1/63 + 1/945.
So the probability of no grandchild making consecutive tweets is
1 - (1 - 4/9 + 1/9 - 1/63 + 1/945) = 4/9 - 1/9 + 1/63 - 1/945 = 47/135.
To find the probability that no consecutive pair of tweets in the email are by the same person, let's consider the number of ways this condition can be satisfied and divide it by the total number of possible outcomes.
Let's focus on the first grandchild's tweets. They make two tweets in a day, so there are 2 possible time slots for each tweet. Additionally, we have four other grandchildren who also make two tweets each, which means we have a total of 10 time slots to fill with these ten tweets.
To satisfy the condition that no consecutive pair of tweets are by the same person, we can use a counting method called "Stars and Bars" or the "Ball and Urn" technique. This method involves representing the time slots as "stars" (*) and using "bars" (|) to separate the time slots for each grandchild.
To ensure no consecutive pair of tweets are by the same person, we need to distribute the 10 time slots among the 5 grandchildren such that each grandchild gets at least one time slot (represented by at least one star (*) between each pair of bars (|)).
Using this technique, we can find the number of ways to satisfy the condition of interest. Let's denote each tweet as a star (*), and the separator between each grandchild's tweets as a bar (|).
For example, one possible arrangement could be:
* | * * | * * * | * | * *
This represents the distribution of time slots among the five grandchildren, where each grandchild makes 2 tweets (represented by the number of stars between the bars).
To find the total number of possible arrangements, we need to consider the number of ways to distribute the stars and bars. The number of ways to distribute the 10 time slots among the 5 grandchildren is given by:
C(10+5-1, 5-1) = C(14, 4)
where C(n, r) represents the binomial coefficient, also known as "n choose r." This formula returns the number of ways to choose r items from a set of n items.
Therefore, the total number of possible outcomes is C(14, 4).
Now, let's calculate the probability by dividing the number of favorable outcomes (those satisfying the condition) by the total number of possible outcomes.
The number of favorable outcomes is the number of arrangements where no consecutive pair of tweets are by the same person. To determine this, we need to consider the distribution of time slots such that each grandchild gets at least one time slot, and no two consecutive stars (*) are allocated to the same grandchild.
One way to approach this is by considering the five grandchildren as a group and arranging the time slots for this group. Then, we need to divide the time slots among the five grandchildren to ensure that each grandchild gets at least one and no two consecutive stars belong to the same grandchild.
By using the same "Stars and Bars" technique, we can find the number of arrangements satisfying this condition:
C(10-5+5-1, 5-1) = C(9, 4)
where 10-5 represents the remaining time slots after allocating one time slot to each grandchild (10 total slots minus 5 grandchildren), and the remaining calculation follows a similar pattern to above.
Thus, the number of favorable outcomes is C(9, 4).
Finally, we can calculate the probability as:
P = C(9, 4) / C(14, 4)
= 126 / 1001
= 18 / 143
Therefore, the value of a + b = 18 + 143 = 161.
So, the value of a + b is 161.