Factorial notation

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How many ways are there to arrange 4 different books side by side on a bookshelf?
factorial notation: n! = n (n-1) (n-2) (n-3) (n-4) . . . . . (5) (4) (3) (2) (1)
by definition : 0! = 1
1. Evaluate: 5!
2. Evaluate: $\frac{{7!}}{{5!}}$
3. Simplify: $\frac{{\left( {n + 3} \right)!}}{{n!}}$
4. Simplify: $\frac{{\left( {n - 1} \right)!}}{{\left( {n + 2} \right)!}}$
5. Simplify: $\frac{{\left( {n + 1} \right)!\;\;\left( {n - 3} \right)!}}{{{{(n!)}^2}}}$
6. Solve: $\frac{{n!}}{{\left( {n - 2} \right)!\;\;3!}} = 7$
arrangement of words "without repititions" = n!
Determine the number of different arrangements of all the letters in the following words:
1. DOG
2. MATH
3. COMPUTER
arrangement of words "with repititions" = $\frac{{n!}}{{\left( {{1^{st}}\;repetition} \right)!\;\;\;\left( {{2^{nd}}\;repetition} \right)!\;\;\;\left( {{3^{rd}}\;repetition} \right)!\; \ldots ..}}$ $\frac{n !}{( 1 ^{s t} re p e t i t i o n ) ! ( 2 ^{n d} re p e t i t i o n ) ! ( 3 ^{r d} re p e t i t i o n ) ! \dots ..}$
Determine the number of different arrangements of all the letters in the following words:
1. ABC vs ABB
2. CANADA
3. BANANA
4. REPETITION
arrangement with restrictions: must deal with the restrictions first!
1. Performing arrangement with restrictions.
  (i) If there are no restrictions
  (ii) If there are restrictions
2. Determine the number of different arrangements of all the letters in the word: COMPUTER
  (i) if there are no restrictions
  (ii) if the vowels must be together
  (iii) the vowels must not be together
3. i) How many ways are there to arrange 3 Math books (Math 10, Math 11, Math 12), 2 physics books (Phys 11, Phys 12), and 5 English (Eng 8, Eng 9, Eng 10, Eng 11, Eng 12) on a bookshelf?
  ii) What if the books on each subject must be kept together?
seating arrangement
1. How many ways can 4 girls and 4 boys sit in a row, if:
  i) they can sit anywhere?
  ii) all the girls must sit together, and all the boys must sit together?
  iii) all the girls must sit together, while the boys can pick their own seats?
  iv) girls and boys alternate?
2. There are 3 couples, and they need to sit together. How many different ways can these 3 couples sit in a row?
3. There are 7 people A, B, C, D, E, F, and G sitting in a row. How many different seating arrangements are there, if:
  i) A must be to the left of B, but they do not need to sit together?
  ii) A and B must sit together?
  iii) A and B cannot sit together?
Seating arrangements are considered to be different only when the positions of the people are different relative to each other.
How many seating arrangements are possible for 7 people sitting around a round table?