Question

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

Survey Reliability A survey with 12 questions is designed so that 3 of the questions are identical and 4 other questions are identical (except for minor changes in wording). How many different ways can the 12 questions be arranged?

Short answer

The number of ways in which the 12 questions can be arranged is equal to 3326400.

Step-by-step solution

Given information

Out of 12 questions, three questions are alike, and four other questions are alike.

Permutation in case of identical units

When out of n units, someunits are identical to each other, the number of ways in which the n selected units (without replacement) can be arranged is given by the following formula:

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

Here, $n_1$units are identical, another set of $n_2$units are identical, and so on until k.

Calculation

It is given that out of 12 questions, one set of four questions are alike, and another set of three questions are alike.

Here,

$$\begin{gathered} n=12 \\ k=2 \\ {n}_1=3 \\ {n}_2=4 \end{gathered}$$

The number of ways in which the 12 questions can be arranged is equal to the following:

$$\begin{gathered} \frac{n!}{n_1!n_2!....n_k!}=\frac{12!}{3!4!} \\ =3326400 \end{gathered}$$

Therefore, the number of ways in which the 12 questions can be arranged is equal to 3326400.