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

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

Step 1: Count the total number of letters in the word 'SCHOOL'. There are 6 letters: S, C, H, O, O, L.
Step 2: Identify if any letters are repeated. The letter 'O' 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 letters!)
Step 4: Calculate the total arrangements. Here, it is 6! (factorial of 6) divided by 2! (factorial of 2 for the repeated 'O').
Step 5: Calculate 6! which is 6 x 5 x 4 x 3 x 2 x 1 = 720.
Step 6: Calculate 2! which is 2 x 1 = 2.
Step 7: Divide the total arrangements by the arrangements of the repeated letters: 720 / 2 = 360.
Step 8: Conclude that there are 360 different ways to arrange the letters of the word 'SCHOOL'.

Permutations of Multisets – The concept of arranging letters where some letters are repeated, requiring the use of factorial division to account for indistinguishable arrangements.