Three people, A, B and C, enter a race.a. Write down the different orders in which they can finish first, second and third. Harry says that there are 3 possible winners, but then only 2 possibilities for second place and only one person left for third place. Is Harry correct? Explain your answer. b. How many different ways can people finish in a 4-person race, 6-person race, and 10-person race?

There will be 6 different orders in which the three people can finish first, second and third. Since, the race consists of three people then we are going to resolve the solution using permutation of the 3 people.

Therefore, we will have

[tex]3!=3\times2\times1=6[/tex]

a) The different orders in which they can finish first, second and third are,

A B C

A C B

B A C

B C A

C A B

C B A

Yes, Harry is correct.

There will be 3 possible winners, only 2 possibilities for second place and only one person left for third place. This can be explained further using the permutation of the three people below,

[tex]\begin{gathered} 3!=3\times2\times1 \\ 3\text{ represents the thr}ee\text{ possible winners} \\ 2\text{ reprsents the two possible winners} \\ 1\text{ represents the only person left for the third place.} \end{gathered}[/tex]

b) The number of ways people can finish in the following was will be,

i) 4-person race

[tex]4!=4\times3\times2\times1=24ways[/tex]

ii) 6-person race

[tex]6!=6\times5\times4\times3\times2\times1=720ways[/tex]

iii) 10-person race

[tex]10!=10\times9\times8\times7\times6\times5\times4\times3\times2\times1=3628800ways[/tex]