Question

In how many ways can the letter of the word "POTATO" be arranged?

• A. 180
• B. 60
• C. 30
• D. 480

Hint:

Always identify and list repeating elements first. For any word arrangement, divide the total factorial of the length by the product of the factorials of the repeating letter counts to avoid overcounting.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem belongs to permutations of multisets, where some elements are identical.
When arranging letters to form distinct words, the repetition of identical letters reduces the total number of unique arrangements.
Key Formula or Approach:
The number of distinct permutations of $n$ objects, where $p_1$ objects are of one type, $p_2$ of another type, and so on, is: \[ \text{Total Arrangements} = \frac{n!}{p_1! \cdot p_2! \cdot \dots \cdot p_k!} \]

Step 2: Detailed Explanation:
Let us analyze the letters in the word "POTATO":
- Total number of letters ($n$) = $6$
Now, let's identify the frequency of each letter:
- Letter 'P' appears: $1$ time
- Letter 'O' appears: $2$ times
- Letter 'T' appears: $2$ times
- Letter 'A' appears: $1$ time
Substitute these counts into our permutation formula: \[ \text{Total Arrangements} = \frac{6!}{1! \cdot 2! \cdot 2! \cdot 1!} \] Calculate the factorials: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \] \[ 2! = 2 \times 1 = 2 \] Substitute these values back: \[ \text{Total Arrangements} = \frac{720}{2 \cdot 2} = \frac{720}{4} = 180 \]

Step 3: Final Answer:
The letters of the word "POTATO" can be arranged in 180 distinct ways.