How many ways can the letters of the word 'BANANA' be arranged?

How many ways can the letters of the word 'BANANA' be arranged?

Correct Answer: 20

Step 1: Count the total number of letters in the word 'BANANA'. There are 6 letters.
Step 2: Identify how many times each letter appears. The letter 'A' appears 3 times, 'B' appears 1 time, and 'N' appears 2 times.
Step 3: Use the formula for arrangements of letters where some letters are repeated. The formula is: Total arrangements = Total letters! / (Repeated letters1! * Repeated letters2! * ...)
Step 4: Plug in the values: Total arrangements = 6! / (3! * 1! * 2!)
Step 5: Calculate 6! which is 720.
Step 6: Calculate 3! which is 6, 1! which is 1, and 2! which is 2.
Step 7: Multiply the factorials of the repeated letters: 3! * 1! * 2! = 6 * 1 * 2 = 12.
Step 8: Divide the total arrangements by the product of the repeated letters: 720 / 12 = 60.
Step 9: The final answer is 60, which is the number of ways to arrange the letters of the word 'BANANA'.

Permutations of Multisets – The arrangement of letters in a word where some letters are repeated is calculated using the formula n! / (n1! * n2! * ... * nk!), where n is the total number of letters and n1, n2, ..., nk are the frequencies of the repeated letters.