Błędnie: There was no solution of the problem at hand, so he decided to take his chance. Poprawnie: There was no solution to the problem at hand, so he decided to take his chance.
Słowo ‘solution’ oznacza “rozwiązanie (zagadnienia, problemu)” i łączy się z przyimkiem ‘to’, np. ‘They could not find any solution to their problem’ (“Nie mogli znaleźć żadnego rozwiązania dla ich problemu”).