1. We are given A=(x,y):2x+3y=23, x,y\in\mathbb N and B= x:(x,y)\in A. We need the number of one-one (injective) functions from $A$ to $B$.

2. First, find all natural number solutions of $2x+3y=23.$

Since $2x=23-3y,$ for $x$ to be a natural number, $23-3y$ must be positive and even.

3. Check possible natural values of $y$. Because $3y<23$, we get $y=1,2,3,4,5,6,7.$ Now test parity:

• If $y=1$, then $x=(23-3)/2=10$ valid.
• If $y=2$, then $x=(23-6)/2=17/2$ not valid.
• If $y=3$, then $x=(23-9)/2=7$ valid.
• If $y=4$, then $x=(23-12)/2=11/2$ not valid.
• If $y=5$, then $x=(23-15)/2=4$ valid.
• If $y=6$, then $x=(23-18)/2=5/2$ not valid.
• If $y=7$, then $x=(23-21)/2=1$ valid.

So, $A=\{(10,1),(7,3),(4,5),(1,7)\}.$ Hence, $|A|=4.$

4. Now form the set $B$ of first coordinates: $B=\{10,7,4,1\}.$ Thus, $|B|=4.$

5. The number of one-one functions from a set with 4 elements to another set with 4 elements is the number of injective maps from a 4-element set to a 4-element set.

Since domain and codomain have equal size, every injective function is a bijection. Hence the number of such functions is $4!=24.$

6. Therefore, the required number is $\boxed{24}.$

7. Comparison with stored answer: Stored correct answer = 24, which matches our result.