Consider a hash table $P[0, 1, \dots, 10]$ that is initially empty. The hash table is maintained using open addressing with linear probing. The hash function used is $h(x) = (x + 7) \bmod 11$. Consider the following sequence of insertions performed on $P$: $$1, 13, 22, 15, 11, 24$$ Which of the following positions in the hash table is/are empty after these insertions are performed?

The hash table has $11$ slots indexed from $0$ to $10$. The hash function is:

Let us trace the insertions step by step:

1. Insert 1:

   $$h(1) = (1 + 7) \bmod 11 = 8$$

   Slot $8$ is empty $\implies P[8] = 1$.

2. Insert 13:

   $$h(13) = (13 + 7) \bmod 11 = 20 \bmod 11 = 9$$

   Slot $9$ is empty $\implies P[9] = 13$.

3. Insert 22:

   $$h(22) = (22 + 7) \bmod 11 = 29 \bmod 11 = 7$$

   Slot $7$ is empty $\implies P[7] = 22$.

4. Insert 15:

   $$h(15) = (15 + 7) \bmod 11 = 22 \bmod 11 = 0$$

   Slot $0$ is empty $\implies P[0] = 15$.

5. Insert 11:

   $$h(11) = (11 + 7) \bmod 11 = 18 \bmod 11 = 7$$

   • Slot $7$ is occupied ($22$).

   • Slot $8$ is occupied ($1$).

   • Slot $9$ is occupied ($13$).

   • Slot $10$ is empty $\implies P[10] = 11$.

6. Insert 24:

   $$h(24) = (24 + 7) \bmod 11 = 31 \bmod 11 = 9$$

   • Slot $9$ is occupied ($13$).

   • Slot $10$ is occupied ($11$).

   • Slot $0$ is occupied ($15$).

   • Slot $1$ is empty $\implies P[1] = 24$.

After all insertions, the occupied positions are:

• Position $0$: $15$

• Position $1$: $24$

• Position $7$: $22$

• Position $8$: $1$

• Position $9$: $13$

• Position $10$: $11$

The empty positions are $2, 3, 4, 5,$ and $6$.

Among the given options, only position 2 (Option C) is empty.