Ways to give gifts to Redhead 
Class has 15 students, of which 6 redheads. Give five awards, while two are intended only to redheads and the remaining 3 are approved for to all students, including redheads. But any student must not receive more than one prize . how many choices?

I give two awards to six redheads
$$\binom{6}{2}$$
I give the remaining 3 prizes to all students, except for two redheads who have already received prizes
$$\binom{13}{3}$$
Total options
$$N = \binom{6}{2} * \binom{13}{3}=4290$$
My answer is not compatible with an answer in the book, which in 2121
Where am I wrong?

You are considering that $2$ prizes are different from the other $3$. If all prizes are the same, just that $2$ of them at least must go to redheads, we can count the number of ways the $5$ prizes can be distributed to all $15$ students and subtract distributions which give $0$ or $1$ prizes to the redheads:
$$N={15\choose5}-{6\choose0}{9\choose5}-{6\choose1}{9\choose4}=3003-126-756=2121$$
