Form a 26-letter word by randomly permuting the 26 letters of the English alphabet. We say that this word is happy if there exists at least one increasing subsequence or decreasing subsequence with length at least 6. If no such subsequence exists, this word is unhappy. For example, the word ALGORITHMSUNDECKBFJPQVWXYZ is happy. Notice that this word has a longest increasing subsequence of length 12 (A,G,I,M,N,P,Q,V,W,X,Y,Z) as well as a longest decreasing subsequence of length 6 (L,I,H,D,C,B). Determine the probability that a randomly-chosen word is happy. NOTE: For each of questions (5), (7), (8), (10), you may provide your answer as a fraction, a percentage, or as a decimal. The choice is yours. Regardless of which option you choose, you will get full credit for an answer that is within 1% of the correct probability.
Form a 26-letter word by randomly permuting the 26 letters of the English alphabet. We say that this word is happy if there exists at least one increasing subsequence or decreasing subsequence with length at least 6. If no such subsequence exists, this word is unhappy.
For example, the word ALGORITHMSUNDECKBFJPQVWXYZ is happy. Notice that this word has a longest increasing subsequence of length 12 (A,G,I,M,N,P,Q,V,W,X,Y,Z) as well as a longest decreasing subsequence of length 6 (L,I,H,D,C,B).
Determine the probability that a randomly-chosen word is happy.
NOTE: For each of questions (5), (7), (8), (10), you may provide your answer as a fraction, a percentage, or as a decimal. The choice is yours. Regardless of which option you choose, you will get full credit for an answer that is within 1% of the correct probability.
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