Suppose that an arrangement is chosen randomly from the set of all arrangements of the eight letters in the word OLYMPICS. What is the probability that the letters O, L, Y and M are not in their original positions in the selected arrangement

To solve the problem, we need to calculate the probability that the letters O, L, Y, and M are not in their original positions in a random arrangement of the eight letters in the word "OLYMPICS."

### Step 1: Calculate the total number of possible arrangements
The word "OLYMPICS" consists of 8 distinct letters. Therefore, the total number of possible arrangements of these letters is:
\[
8! = 40320
\]

### Step 2: Determine the number of favorable arrangements
The problem requires that the letters O, L, Y, and M are not in their original positions. This is an example of a derangement problem (a permutation where none of the elements appear in their original position) for the letters O, L, Y, and M.

#### Step 2a: Calculate the number of derangements of 4 letters
The number of derangements of \( n \) objects (denoted by \( !n \)) can be calculated using the following formula:
\[
!n = n! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^n}{n!} \right)
\]

For \( n = 4 \), the number of derangements \( !4 \) is:
\[
!4 = 4! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} \right)
\]
\[
= 24 \left( 1 - 1 + 0.5 - 0.1667 + 0.0417 \right)
\]
\[
= 24 \times 0.375 = 9
\]

So, there are 9 ways to derange the letters O, L, Y, and M.

### Step 3: Calculate the number of favorable arrangements
Since the letters O, L, Y, and M can be deranged in 9 ways, and the remaining letters (P, I, C, S) can be arranged in any order, we have:
\[
9 \times 4! = 9 \times 24 = 216
\]
favorable arrangements where O, L, Y, and M are not in their original positions.

### Step 4: Calculate the probability
The probability that the letters O, L, Y, and M are not in their original positions is given by:
\[
P = \frac{\text{Number of favorable arrangements}}{\text{Total number of arrangements}} = \frac{216}{40320} = \frac{1}{186.67}
\]

### Final Answer
The probability that the letters O, L, Y, and M are not in their original positions is approximately:
\[
\boxed{\frac{1}{186.67}}
\]

This is a precise answer, but you may also approximate it as:
\[
\boxed{0.00536}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What is the probability that all letters are in their original positions?
2. How many arrangements of the word OLYMPICS start with the letter O?
3. What is the probability that exactly one of the letters O, L, Y, or M is in its original position?
4. How many distinct arrangements can be made if one letter in "OLYMPICS" is repeated?
5. What is the probability that the first and last letters in the arrangement are consonants?

**Tip:** In permutation problems, identifying derangements can simplify solving for specific scenarios where elements must be out of their original positions.

Calculate the probability that the letters O, L, Y, and M are not in their original positions in a random arrangement of the word OLYMPICS. Understand derangements and permutations with a detailed solution. Suitable for advanced high school students and those interested in probability problems.

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