Mrs. Crenshaw has confiscated 4 cell phones from 4 different students because they used them during class. To get the cell phones back, they have to answer the following question: What is the probability that no student receives his or her own phone if Mrs. Crenshaw hands them back at random at the end of the school day? What answer should they give? Explain.

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To find the probability that no student receives his or her own phone, we need to consider the number of ways the cell phones can be handed back to the students.

Let's break down the problem step by step:

Step 1: Calculate the total number of ways the cell phones can be handed back.
Since each student has a cell phone already confiscated, there are 4 different phones available to hand back to the 4 students. The number of ways to distribute these phones is given by the factorial of the number of students. In this case, the factorial of 4 is 4! = 4 x 3 x 2 x 1 = 24.

Step 2: Calculate the number of ways no student receives his or her own phone.
For none of the students to receive their own phone, it means that all students should get someone else's phone. This is similar to solving the derangement problem. The number of derangements is given by the nearest integer value to the ratio of the factorial of the number of students to the constant value e (approximately 2.71828). In this case, we have 4 students, so the derangement is calculated as int(4! / e) = int(24 / 2.71828) = int(8.82918) = 9.

Step 3: Calculate the probability.
The probability of no student receiving his or her own phone is the number of ways no student receives their own phone divided by the total number of ways to distribute the phones. Therefore, the probability is 9 / 24 = 3 / 8.

Therefore, the answer Mrs. Crenshaw should be given is that the probability of no student receiving his or her own phone when the cell phones are handed back at random is 3/8.