Secret Partner Choices

The number of ways in which each student can choose a partner is $3^{4}$ . The number of ways in which we can pair four people is $3$ as they are:
1. $(AB) (CD)$
2. $(AC) (BD)$
3. $(AD) (BC)$ .

So $P=\frac{1}{27}$ and the answer is $27$ .

A chooses B, B chooses A ;C chooses D ; D chooses C. That is one favorable outcome. Now think of A.C and B.D similiarly then think of A.D and B.C So Number of favorable outcomes is 3. Number of all possible outcomes is 3 3 3*3 that is 3^4=81 hint: A can choose anyone out of remaining 3 guys and similiarly B, C and D can do.

so the required probability, p=3/81=1/27 so 1/p= 27

Since each student can choose one of the other three, the total number of choices is $3^4 = 81$ . For everyone to choose a partner who chose him/her, there are $3$ ways of splitting the $4$ students in two separate groups. Thus the probability is $\frac{3}{81}=\frac{1}{27}$ . Thus $\frac1p=27$ .

If we will count the ways, there are only s=3 cases that every student chooses a partner who chose him/ her. Each student can choose only one from the other three students. So, the total number of cases in which each student selects his/her group partner is t=(3)(3)(3)(3)=81. Therefore, the probability then is P=s/t=3/81=1/27, and thus, the reciprocal of P is 27.

Let us denote the students by S1, S2, S3, and S4. Let us further assume that S1 picks S2 (this choice is arbitrary; S1 could just as well choose S3 or S4 and we would get the same result).

Now, because S1 picked S2, S2 must pick S1. The probability of this happening is 1/3 because S2 has three other students to choose from.

Now that S1 and S2 are paired up, we see that S3 and S4 must choose each other. The probability of S3 choosing S4 is 1/3, and the probability of S4 choosing S3 is 1/3.

The total probability P is then the product of the three individual probabilities:

P = \frac {1}{3} * \frac {1}{3} * \frac{1}{3}

= \frac {1}{27}

And hence \frac {1]{P} = 27.

Each person has $3$ choices for his/her partner, so there are a total of $3^4$ outcomes. Of which, there are only $3$ ways for everyone to chose a partner who chose him/her. Hence $P = \frac {3}{3^4} = \frac {1}{27}$ . Hence, $\frac {1}{P} = 27$ .