The number of onto functions (surjective functions) from set X = {1,2,3,4} to set Y = {a,b,c} is ________.

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Accepted Answer

36 We want functions from 4 inputs onto 3 outputs that hit every one of a, b, c. There are 3^4 = 81 functions total, but many miss a target value. Inclusion-exclusion removes the maps that skip at least one value: subtract the 3 ways to drop a value (each leaving 2^4 maps) and add back the 3 ways to drop two values (each leaving 1^4). That gives 81 - 48 + 3 = 36.