How many different ways can the letters of the word 'BOOK' be arranged? (2014)

How many different ways can the letters of the word 'BOOK' be arranged? (2014)

Step 1: Identify the letters in the word 'BOOK'. The letters are B, O, O, and K.
Step 2: Count the total number of letters. There are 4 letters in total.
Step 3: Notice that the letter 'O' appears twice. This means we have repeating letters.
Step 4: To find the number of different arrangements, we use the formula for permutations of multiset: total letters factorial divided by the factorial of the number of times each letter repeats.
Step 5: Calculate the total arrangements using the formula: 4! (factorial of total letters) divided by 2! (factorial of the repeating letter 'O').
Step 6: Calculate 4! = 4 × 3 × 2 × 1 = 24.
Step 7: Calculate 2! = 2 × 1 = 2.
Step 8: Divide the total arrangements by the arrangements of the repeating letters: 24 / 2 = 12.
Step 9: Therefore, the number of different ways to arrange the letters of the word 'BOOK' is 12.

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