Question

You and six friends play a game where each person writes down his or her name on a scrap of paper, and the names are randomly distributed back to each person. Find the probability that everyone gets back his or her own name.

Answer 1

Answer with explanation:

Total number of different candidates who are playing the game=7

Suppose, Seven candidates are represented  by ={A,B,C,D,E,F,G}

Total Possible Outcome =7

→Probability that , "A" gets his scrap of paper , means the paper on which he or she has written his or her name

                             $=\frac{\text{total favorable outcome}}{\text{total favorable outcome}}\\\\=\frac{1}{7}$

→Now, 6 candidates are left.

Probability that , "B" gets his scrap of paper , means the paper on which he  or she has written his or her name

                             $=\frac{\text{total favorable outcome}}{\text{total favorable outcome}}\\\\=\frac{1}{6}$  

→Now, 5, candidates are left.

Probability that , "C" gets his scrap of paper , means the paper on which he or she has written his or her name

                             $=\frac{\text{total favorable outcome}}{\text{total favorable outcome}}\\\\=\frac{1}{5}$  

→Now, 4 candidates are left.

Probability that , "D" gets his scrap of paper , means the paper on which he or she has written his or her name

                             $=\frac{\text{total favorable outcome}}{\text{total favorable outcome}}\\\\=\frac{1}{4}$  

→Now, 3 candidates are left.

Probability that , "E" gets his scrap of paper , means the paper on which he or she has written his or her name

                             $=\frac{\text{total favorable outcome}}{\text{total favorable outcome}}\\\\=\frac{1}{3}$  

→Now, 2 candidates are left.

Probability that , "F" gets his scrap of paper , means the paper on which he or she has written his or her name

                             $=\frac{\text{total favorable outcome}}{\text{total favorable outcome}}\\\\=\frac{1}{2}$  

→Now, a single candidates is left.

Probability that , "G" gets his scrap of paper , means the paper on which he or she has written his or her name

                             $=\frac{\text{total favorable outcome}}{\text{total favorable outcome}}\\\\=\frac{1}{1}=1$  

Required Probability

                 $=\frac{1}{7} \times\frac{1}{6} \times\frac{1}{5} \times\frac{1}{4} \times\frac{1}{3} \times\frac{1}{2} \times 1\\\\=\frac{1}{5040}$    

   

Answer 2

This  will be

1/6 * 1/5*1/4*1/3*1/2 =  1/720