Your statistics instructor administers a test having five multiple-choice items with four options each. List the ways in which one can guess correctly on exactly four items on this test.

Are there 24 ways, because: 4!
(4)(3)(2)(1) =24?

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Yes, you are correct. There are indeed 24 ways to guess correctly on exactly four items on this test. Here's how you can calculate it:

To solve this problem, we can consider the five multiple-choice items individually.

For each item, there are four options to choose from. In order to guess correctly on exactly four items, we need to choose one incorrect option and three correct options.

Let's break this down step by step:

For the first item, we need to choose one incorrect option out of the four available. This can be done in 4 ways.

For the second item, we again need to choose one incorrect option out of the remaining three options (since we have already chosen one incorrect option for the first item). This can be done in 3 ways.

For the third item, we need to choose one incorrect option out of the remaining two options. This can be done in 2 ways.

For the fourth item, we have only one option left to choose from, which will be the incorrect option.

Finally, for the fifth item, there is no choice to be made since all options have already been determined.

To find the total number of ways to guess correctly on exactly four items, we multiply the number of choices for each item together:

4 options for the first item * 3 options for the second item * 2 options for the third item * 1 option for the fourth item = 4 * 3 * 2 * 1 = 24

Therefore, there are 24 ways to guess correctly on exactly four items on this test.