A $\emph{planar}$ graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let $G$ be a planar graph with $11$ vertices of degree $2$, $5$ vertices of degree $3$, and $1$ vertex of degree $7$. Find the number of faces into which $G$ divides the sphere.
1. **Identify the number of vertices \( V \):**
   The graph \( G \) has:
   - 11 vertices of degree 2,
   - 5 vertices of degree 3,
   - 1 vertex of degree 7.
   
   Therefore, the total number of vertices \( V \) is:
   \[
   V = 11 + 5 + 1 = 17
   \]

2. **Calculate the number of edges \( E \):**
   Using the degree sum formula, which states that the sum of the degrees of all vertices is equal to twice the number of edges:
   \[
   \sum \text{degree}(v) = 2E
   \]
   The sum of the degrees of the vertices is:
   \[
   2 \times 11 + 3 \times 5 + 7 \times 1 = 22 + 15 + 7 = 44
   \]
   Therefore, the number of edges \( E \) is:
   \[
   2E = 44 \implies E = \frac{44}{2} = 22
   \]

3. **Use Euler's formula for planar graphs:**
   Euler's formula for a connected planar graph states:
   \[
   V - E + F = 2
   \]
   where \( F \) is the number of faces. Substituting the known values:
   \[
   17 - 22 + F = 2
   \]
   Solving for \( F \):
   \[
   F = 2 + 22 - 17 = 7
   \]

Conclusion:
\[
\boxed{7}
\]