Ex. 13
Determine the number of ways in which the letters of the word ANDTHEREAFTER can be arranged.

Step 1 ：First, count the occurrences of each letter in the word ANDTHEREAFTER: \(A: 2, N: 1, D: 1, T: 2, H: 1, E: 3, R: 2, F: 1\)

Step 2 ：Use the formula for permutations with repetitions: \(\frac{n!}{a!b!c!...}\), where n is the total number of letters and a, b, c, ... are the occurrences of each letter

Step 3 ：Calculate the factorial of the total number of letters: \(n! = 13! = 6,227,020,800\)

Step 4 ：Calculate the factorial of the occurrences of each letter: \(2! = 2, 1! = 1, 3! = 6\)

Step 5 ：Calculate the denominator: \(2! \times 1! \times 1! \times 2! \times 1! \times 3! \times 2! \times 1! = 48\)

Step 6 ：Divide the numerator by the denominator: \(\frac{6,227,020,800}{48} = 129,729,600\)

Step 7 ：\(\boxed{129,729,600}\) ways to arrange the letters of the word ANDTHEREAFTER