Four students (A, B, C, D) are assigned to four projects (P1, P2, P3, P4), one each.
- A does not get P1.
- B gets P2 or P3.
- C does not get P3.
- D gets P4.
- Step 1: Apply conditions. D gets P4. B gets P2 or P3. A does not get P1. C does not get P3.
- Step 2: Assign D. D = P4. Remaining: P1, P2, P3 for A, B, C.
- Step 3: Assign B. B gets P2 or P3.
- Step 4: Assign others. A not P1, so A gets P2 or P3. C not P3, so C gets P1 or P2.
Try B = P2: A, C get P1, P3. C not P3, so C = P1, A = P3.
Arrangement: C (P1), B (P2), A (P3), D (P4).
Valid. Try B = P3: A, C get P1, P2. C = P1, A = P2 (A not P1). Valid.
- Step 5: Check question. B gets P2 or P3. Options suggest one answer.
From arrangement, B = P2 is consistent.
- Step 6: Check options. Options: (1) P1, (2) P2, (3) P3, (4) P4. P2 matches option (2).
- Step 7: Conclusion. Option (2) is correct.
When $10^{100}$ is divided by 7, the remainder is ?