help me to correct my exercise on nested quantifier

Let F(x,y) be the statement "x can fool y", where the domain consists of all people in the world. Use quantifiers to express each of these statements.

a. Nancy can fool exactly two people

my answer is:

$\displaystyle \exists x\exists y\forall z [ (F(Nancy, x) \wedge F(Nancy, y)) \wedge ((z = x) \wedge (z = y))]$

b. No one can fool himself or herself

my answer is:

$\displaystyle \exists x\forall y [\neg F(x,x) \wedge ((y=x))]$

c. There is exactly one person whom everybody can fool

my answer is:

$\displaystyle \exists x\forall z [F(z,x) \wedge (x \neq z)]$

Are my answer's correct?