There are 13!/(3!^4) ways of AAA occurring.

Same for BBB, CCC, DDD, EEE.   Totaling 5*13!/(3!^4) = 24024000

There are 11!/(3!^3) ways of AAA and BBB.

Same for all C(5,2)= 10 double triples. Total is 10*11!/(3!^3) = 1848000

There are 9!/(3!^2) ways of AAA and BBB and CCC

Same for all C(5,3)= 10 triple triples. Total is 10*9!/(3!^2) =  100800

There are 7!/(3!) ways of AAA and BBB and CCC and DDD

Same for all C(5,4)= 5 quad triples. Total is 5*7!/(3!) = 4200

There are 5! ways of getting all of AAA, BBB, CCC, DDD and EEE. = 120

By inclusion/exclusion, ways of getting at least 1 triple = 24024000 - 1848000 + 100800 - 4200 + 120 = 22,272,720

Since we want no triples, the number of ways of that is 15!/3!^5 - 22272720 =  145,895,280.

In a million trials we'd expect around 132443 cases of finding at least one triple letter sequence. That way we can test the result with a simulation:

a='aaabbbcccdddeee';

ct=0;

for i=1:1000000

  b=a(randperm(15));

  found=false;

  for j=1:3:13

    f=strfind(b,a(j:j+2));

    if ~isempty(f)

      found=true;

    end

  end

  ct=ct+found;

end

fprintf('%d vs 132443',ct)

Three runs show

133132 vs 132443

132654 vs 132443

131935 vs 132443

a pretty good match.

A 10-million-trial run shows 1323536 vs 1324433.