Question

Exercise 3: Find the number of words a1a2... ag of length 9 such that at least 4 consec- utive letters are the same? (In other terms, for each i e{1,...,9), a, E fa, b, c,..., z, and there exists an integer l < j < 6 such that aj-dj+,-dj+2-aj+3.)

Answer 1

Let such a bunch of 4 consecutive letters start at index k

The possibilities are k = 1, 2, 3, 4, 5, 6 but not k = 7, 8, 9 (since if a bunch of size 4 starts at k=7 it can go only up to k=9)

If it starts at k=1, then the number of such letters is as the bunch (which is the bunch of same letters) can be chosen in 26 ways. For each of the other 5 letters, there are 26 possibilities

26.26 5

If it starts at k=2 then the first letter must be different from the bunch so we have possible numbers

25 26 264

If it starts at k=3 then we have and so on till k=6

26.25 264

So the total number of such letters is

26 2655.26 25 264- 1794087776

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