Question

Consider two sets $A = \{2, 3, 5, 7, 11, 13\}$ and $B = \{1, 8, 27\}$. Let $f$ be a function from $A$ to $B$ such that for every element $b$ in $B$, there is at least one element $a$ in $A$ such that $f(a) = b$. Then, the total number of such functions $f$ is

Accepted Answer

A. Number of elements in domain $|A| = 6$ and codomain $|B| = 3$. The condition states that $f$ is an onto (surjective) function. Using the principle of inclusion-exclusion, the number of onto functions is: $$3^6 - \binom{3}{1}2^6 + \binom{3}{2}1^6 = 729 - 3(64) + 3(1) = 729 - 192 + 3 = 540$$

Number of Onto Functions

Explanation

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