A teacher suggests four possible books for students to read. Each of six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student?
To solve this problem, we need to count the number of ways to distribute six students among four books such that each book is read by at least one student. This is a classic problem that can be solved using the principle of inclusion-exclusion.

1. **Total number of ways to distribute six students among four books:**
   Each student has 4 choices, so the total number of ways without any restrictions is:
   \[
   4^6
   \]

2. **Using the principle of inclusion-exclusion:**
   We need to subtract the cases where at least one book is not read by any student.

   - **Case 1: Exclude one book:**
     There are \(\binom{4}{1}\) ways to choose which book is excluded, and the remaining 6 students can choose among the remaining 3 books:
     \[
     \binom{4}{1} \cdot 3^6
     \]

   - **Case 2: Exclude two books:**
     There are \(\binom{4}{2}\) ways to choose which two books are excluded, and the remaining 6 students can choose among the remaining 2 books:
     \[
     \binom{4}{2} \cdot 2^6
     \]

   - **Case 3: Exclude three books:**
     There are \(\binom{4}{3}\) ways to choose which three books are excluded, and the remaining 6 students can choose the remaining 1 book:
     \[
     \binom{4}{3} \cdot 1^6
     \]

   - **Case 4: Exclude all four books:**
     This is not possible since there are no books left to choose from:
     \[
     \binom{4}{4} \cdot 0^6 = 0
     \]

3. **Applying the principle of inclusion-exclusion:**
   \[
   4^6 - \binom{4}{1} \cdot 3^6 + \binom{4}{2} \cdot 2^6 - \binom{4}{3} \cdot 1^6
   \]

4. **Calculating each term:**
   \[
   4^6 = 4096
   \]
   \[
   \binom{4}{1} \cdot 3^6 = 4 \cdot 729 = 2916
   \]
   \[
   \binom{4}{2} \cdot 2^6 = 6 \cdot 64 = 384
   \]
   \[
   \binom{4}{3} \cdot 1^6 = 4 \cdot 1 = 4
   \]

5. **Combining all terms:**
   \[
   4096 - 2916 + 384 - 4 = 1560
   \]

Thus, the number of ways to distribute the six students among the four books such that each book is read by at least one student is:
\[
\boxed{1560}
\]