Maths
posted by John on .
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 email 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 email 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?

9

fail.

First, we will find the probability of at least one grandchild making consecutive tweets by using the inclusionexclusion principle, and then subtract the result from 1.
From the inclusionexclusion 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.