In Exercises 5–36, express all probabilities as fractions.

What a Word! One of the longest words in standard statistics terminology is “homoscedasticity.” How many ways can the letters in that word be arranged?

The letters of the word “homoscedasticity” are to be arranged.

The number of unique ways to arrange n items where few of them are of same type is established using following formula.

The following is its formula:

$\frac{n!}{n_1!\times n_2!\times ....\times n_k!}$

Here, out of a total of n items, $n_1$ are alike, $n_2$ are alike, and it continues until $n_k$ are alike.

The total number of letters is 16.

The letters that are identical or repetitive are “o,” “s,” “c,” “t,” and “i”.

Here,the letters “o,” “s,” “c,” “t,” and “i” are all repeating twice.

The number of ways in which the 16 letters can be arranged is determined below:

$$\begin{gathered} \frac{n!}{n_1!\times n_2!\times ....\times n_k!}=\frac{16!}{2! \times 2!\times 2!\times 2!\times 2!} \\ =653,837,184,000 \end{gathered}$$

Therefore, the number of unique ways in which the letters of the word “homoscedasticity” can be arranged is 653,837,184,000.