Factorials in a Set

Since the answer is 1 greater than a multiple of 7, it cannot be divisible by 7. Therefore, we only need to check all of them up to 6.

$0!\equiv 1 \bmod{7}$

$1!\equiv 1 \bmod{7}$

$2!\equiv 2*1 \bmod{7} \equiv 2 \bmod{7}$

$3!\equiv 3*2 \bmod{7} \equiv 6 \bmod{7}$

$4!\equiv 4*6 \bmod{7} \equiv 3 \bmod{7}$

$5!\equiv 5*3 \bmod{7} \equiv 1 \bmod{7}$

$6!\equiv 6*1 \bmod{7} \equiv 6 \bmod{7}$

Therefore, the only ones that work are 0, 1, and 5. So the answer is $\boxed{3}$ .