To find the number of ways to reorder the letters of the word "roblox" in at least some new ways, we first consider the total number of possible arrangements of the letters, taking into account the repeated letters.

Step 1: Identify the letters and their frequencies

The word "roblox" consists of 6 letters where:

• 'r' appears 1 time.
• 'o' appears 2 times.
• 'b' appears 1 time.
• 'l' appears 1 time.
• 'x' appears 1 time.

Step 2: Calculate the total number of permutations

The total number of permutations of the word "roblox" can be calculated using the formula:

$\text{Total permutations} = \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}$

where $n$ is the total number of letters, and $n_1, n_2, \ldots, n_k$ are the factorials of the counts of each distinct letter.

For "roblox":

$\text{Total permutations} = \frac{6!}{2!} = \frac{720}{2} = 360$

Step 3: Calculate the number of ways to reorder letters

Since we are asked for the number of ways to reorder the letters "in at least some new orders," this encompasses all distinct permutations except the original arrangement.

The number of permutations excluding the original order:

$\text{New permutations} = 360 - 1 = 359$

Therefore, the number of ways to reorder the letters of "roblox" in at least some new orders is:

$\boxed{359}$

Would you like more details, or do you have any questions?

Related Questions:

1. How many ways can you reorder the word "balloon" considering repeated letters?
2. How do you find permutations of words with more than two repeated letters?
3. What is the general formula for permutations of objects with identical items?
4. How can you calculate permutations when only specific letters are to be reordered?
5. How would the answer change if one more letter was added to "roblox"?

Tip: Always account for repeated letters when calculating permutations to avoid overcounting identical arrangements!