There are 4 students in a class. A teacher wants them to each secretly choose a partner for a group project. If everyone independently choses a

partner randomly, the probability that everyone choses a partner who chose him/her is P. What is the value of 1/P?

To solve this question, we need to find the probability that everyone chooses a partner who chose them.

Let's consider the first student. The probability that the first student chooses a specific partner who chose them is 1/3, as there are three remaining students to choose from.

Now, let's move on to the second student. The probability that the second student chooses a specific partner who chose them is also 1/3, but we need to consider two scenarios:
1) If the first student chose the second student, then the second student needs to choose the first student (probability 1/3).
2) If the first student chose a different student, then the second student needs to choose the remaining student who isn't the first student (probability 1/2).

Now, let's consider the third student. Similar to the second student, there are two scenarios to consider:
1) If both the first and second students chose the third student, then the third student needs to choose one of them (probability 1/3).
2) If either the first or second student did not choose the third student, then the third student needs to choose the remaining student who isn't chosen (probability 1).

Lastly, the fourth student automatically gets the remaining partner.

To find the probability that everyone chooses a partner who chose them, we multiply the probabilities at each step:

Probability = (1/3) * (1/3) * (1/3) * 1 = 1/27

Therefore, the value of 1/P is simply the reciprocal of the probability:

1/P = 1 / (1/27) = **27**

So, the value of 1/P is 27.