Interesting, and startling to me when I first learned of it, is the probabilistic method [1], now not infrequently used in combinatorics to prove the existence of a solution to some question without constructing it. One proves that the probability of existence is strictly greater than zero.

probabilistic method[1], now not infrequently used in combinatorics to prove the existence of a solution to some question without constructing it. One proves that the probability of existence is strictly greater than zero.[1] https://www.amazon.com/Probabilistic-Method-Discrete-Mathema...

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