Calculate the number of distinct arrangements for letters in a word and for distinct items in a set.

The word "BOOK" has 4 letters. The letter 'O' is repeated 2 times.

where $n$ is the total number of letters and $n_1$ is the number of times a letter is repeated.

The given answer is 12 ways.
The answer is $\boxed{\text{12 ways}}$.

2) To find the number of ways 5 books can be arranged on a shelf, we calculate the permutations of 5 distinct items.

There are 5 books.

The given answer is 120 ways.
The answer is $\boxed{\text{120 ways}}$.

3) To find the number of different seating arrangements for 8 people in a row, we calculate the permutations of 8 distinct items.

There are 8 people.

The given answer is 40,320.
The answer is $\boxed{\text{40,320 ways}}$.

4) To find the number of ways the letters in "MANGO" can be arranged if the vowels must always come together, we treat the vowels as a single unit.

The word "MANGO" has letters M, A, N, G, O.
Vowels: A, O. Consonants: M, N, G.
Treat (AO) as one unit. Now we are arranging M, N, G, (AO). This is 4 units.

The 4 units can be arranged in $4!$ ways.

The vowels (A, O) can be arranged among themselves in $2!$ ways.

The given answer is 48 ways.
The answer is $\boxed{\text{48 ways}}$.

5) To find the number of ways the letters in "ENGINEER" can be arranged such that all vowels are together, we treat the vowels as a single unit and account for repetitions.

The word "ENGINEER" has 8 letters: E, N, G, I, N, E, E, R.
Vowels: E, I, E, E (3 E's, 1 I). Consonants: N, G, N, R (2 N's, 1 G, 1 R).
Group the vowels: (EIE E). Treat this as one unit.
Now we are arranging the units: (EIE E), N, G, N, R. This is 5 units.

The units are (Vowel Group), N, G, N, R. There are 5 units in total, with 'N' repeated 2 times.

The vowel group is (EIE E). There are 4 vowels, with 'E' repeated 3 times.

The given answer is 120 ways. My calculation gives 240 ways. Let me recheck.

Recheck for Question 5:
Word: ENGINEER
Total letters: 8
Vowels: E, I, E, E (4 vowels, 3 E's, 1 I)
Consonants: N, G, N, R (4 consonants, 2 N's, 1 G, 1 R)

Treat all vowels as one block: (EIEE).
Now we have 5 "items" to arrange: (EIEE), N, G, N, R.
Number of ways to arrange these 5 items:
There are 5 items, but 'N' is repeated twice.

Number of ways to arrange the letters within the vowel block (EIEE):
There are 4 letters, with 'E' repeated 3 times.

Total number of arrangements = (Arrangement of blocks) $\times$ (Arrangement within vowel block)

The provided answer of 120 ways is incorrect based on my calculation. The correct answer is 240 ways.
The answer is $\boxed{\text{240 ways}}$.

6) To find the number of different ways 4 paintings can be arranged on a wall, we calculate the permutations of 4 distinct items.

There are 4 paintings.

The given answer is 24 ways.
The answer is $\boxed{\text{24 ways}}$.