Let x and y denote the sets containing 4 and 4 distinct objects respectively and f denote the set of all possible functions defined from x and y. let f be randomly chosen from f. what is the probability of f being onto?
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Onto = Total functions - Into
Into = 4C1 x (4-1)⁴ - 4C2 x (4-2)^² + 4C3 x (4-3)⁴ = 424
(Refer: https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://math.stackexchange.com/questions/334420/number-of-onto-functions&ved=2ahUKEwiStdrUoYnZAhUIq48KHdKgC-4QFjAAegQIDxAB&usg=AOvVaw2uM9itDmyhFS52l1UJww0v)
Therefore, Onto = 4⁴ - 424
Prob (onto) = 24/256 = 0.093
Into = 4C1 x (4-1)⁴ - 4C2 x (4-2)^² + 4C3 x (4-3)⁴ = 424
(Refer: https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://math.stackexchange.com/questions/334420/number-of-onto-functions&ved=2ahUKEwiStdrUoYnZAhUIq48KHdKgC-4QFjAAegQIDxAB&usg=AOvVaw2uM9itDmyhFS52l1UJww0v)
Therefore, Onto = 4⁴ - 424
Prob (onto) = 24/256 = 0.093
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