Find the number of permutations of the letters of the word SCHOOL.

Answer Details

To find the number of permutations (different orders) of the letters in the word SCHOOL, you need to know two key concepts:

• Permutation means an arrangement of all items where order matters.
• If some items (letters) are repeated, we must adjust the calculation to account for these repeats so we don’t count identical arrangements multiple times.

The word SCHOOL has 6 letters: S, C, H, O, O, L.

• The letter O is repeated 2 times, while the other letters appear only once.

If all 6 letters were different, the total number of permutations would be:

\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

But because two of the letters (O's) are identical, each arrangement is counted twice (once for each O in each possible position). To correct this, we divide by the number of ways to arrange the two O’s among themselves:

\[ \text{Number of permutations} = \frac{6!}{2!} \]

Now, calculate each part:

• \(6! = 720\)
• \(2! = 2 \times 1 = 2\)

So,

\[ \text{Number of permutations} = \frac{720}{2} = 360 \]

Explanation: We divide by \(2!\) because swapping the two identical O's does not produce a new unique arrangement. For every arrangement of the 6 letters, the O's can be swapped, so we would be counting each unique arrangement twice if we did not divide by \(2!\).

Summary: The number of distinct permutations of the letters of the word SCHOOL is 360.