Question 1210199
Let's solve this problem using the Principle of Inclusion-Exclusion.

**1. Total Number of Sequences**

* Each a_i can be any of the 8 numbers {1, 2, 3, 4, 5, 6, 7, 8}.
* There are 8 positions in the sequence.
* Total number of sequences = 8⁸.

**2. Inclusion-Exclusion Principle**

Let S be the set of all sequences. Let A_i be the set of sequences that do not contain the number i. We want to find |S \setminus (A_1 ∪ A_2 ∪ ... ∪ A_8)|.

* |S| = 8⁸
* |A_i| = 7⁸ (sequences without the number i)
* |A_i ∩ A_j| = 6⁸ (sequences without numbers i and j)
* |A_i ∩ A_j ∩ A_k| = 5⁸ (sequences without numbers i, j, k)
* ...
* |A_1 ∩ A_2 ∩ ... ∩ A_7| = 1⁸ (sequences without numbers 1 to 7)
* |A_1 ∩ A_2 ∩ ... ∩ A_8| = 0⁸ = 0 (sequences without all numbers 1 to 8)

By the Principle of Inclusion-Exclusion:

|A_1 ∪ A_2 ∪ ... ∪ A_8| = Σ |A_i| - Σ |A_i ∩ A_j| + Σ |A_i ∩ A_j ∩ A_k| - ...

|A_1 ∪ A_2 ∪ ... ∪ A_8| = ⁸C₁ * 7⁸ - ⁸C₂ * 6⁸ + ⁸C₃ * 5⁸ - ⁸C₄ * 4⁸ + ⁸C₅ * 3⁸ - ⁸C₆ * 2⁸ + ⁸C₇ * 1⁸ - ⁸C₈ * 0⁸

The number of sequences we want is:

8⁸ - |A_1 ∪ A_2 ∪ ... ∪ A_8|

**3. Calculate the Values**

* 8⁸ = 16777216
* ⁸C₁ = 8
* ⁸C₂ = 28
* ⁸C₃ = 56
* ⁸C₄ = 70
* ⁸C₅ = 56
* ⁸C₆ = 28
* ⁸C₇ = 8
* ⁸C₈ = 1

* 8 * 7⁸ = 8 * 5764801 = 46118408
* 28 * 6⁸ = 28 * 1679616 = 47029248
* 56 * 5⁸ = 56 * 390625 = 21875000
* 70 * 4⁸ = 70 * 65536 = 4587520
* 56 * 3⁸ = 56 * 6561 = 367416
* 28 * 2⁸ = 28 * 256 = 7168
* 8 * 1⁸ = 8 * 1 = 8
* 1 * 0⁸ = 0

* |A_1 ∪ A_2 ∪ ... ∪ A_8| = 46118408 - 47029248 + 21875000 - 4587520 + 367416 - 7168 + 8 - 0
* |A_1 ∪ A_2 ∪ ... ∪ A_8| = 16334520

* 8⁸ - |A_1 ∪ A_2 ∪ ... ∪ A_8| = 16777216 - 16334520 = 442696

**Therefore, the number of sequences is 442696.**