Solution

Given: \( n(A)=6 \) and \( n(B)=3 \).

Formula: The number of onto (surjective) functions from a set of size \(m\) to a set of size \(n\) is \[ n! \, S(m,n) \] where \(S(m,n)\) is the Stirling number of the second kind (number of ways to partition \(m\) elements into \(n\) non-empty subsets).

We can also use the Inclusion–Exclusion Principle: \[ n! \, S(m,n) = \sum_{k=0}^{n} (-1)^k \binom{n}{k}(n-k)^m \] For \(m=6,\ n=3\): \[ N = 3^6 - 3\times 2^6 + 3\times 1^6 \]

Calculation: \[ 3^6 = 729,\quad 2^6 = 64 \] \[ N = 729 - 3(64) + 3(1) = 729 - 192 + 3 = 540. \]

Answer: The number of onto functions is \[ \boxed{540}. \]