Question

In a relay race there are six teams A, B, C, D, E and F. What is the probability that A, B, C, finish first, second, third respectively?
A) $\dfrac{1}{2}$
B) $\dfrac{1}{{12}}$
C) $\dfrac{1}{{60}}$
D) $\dfrac{1}{{120}}$

Answer 1

Hint: First we have to find all the possibilities in which all the teams can finish the race and then after we will have to fix up the teams A, B and C as given, find the total number of teams by adding up.

Complete step-by-step solution:
Given: Six teams are given that is A, B, C, D, E and F and all the teams are participating in a relay race.
Now in this question we have to find the probability that A, B, C, finish first, second, third respectively.
First we will find the probability in which all the teams can finish the relay race.
As one team can require only one position, therefore six teams can finish the race in $6!$ ways (because repetition can’t be possible and hence one team will be no more available for the next position).
Now if we fix the position of team A, team B, team C in finishing the race on first, second and third position respectively then we have to find the no. of possibilities of rest of three teams that can be arranged.
The number of ways in which the positions to the last three teams can be given in $3!$ ways (because only three teams left and hence three positions left).
So the required probability that A, B, C finish first, second, third respectively
$ = \dfrac{{{\text{possibility}}\,{\text{in}}\,{\text{which}}\,{\text{rest}}\,{\text{of}}\,{\text{three}}\,{\text{teams}}\,{\text{can}}\,{\text{be}}\,{\text{arranged}}}}{{{\text{total}}\,{\text{possibilities}}}}$
$
   = \dfrac{{3!}}{{6!}}\, = \,\dfrac{{3!}}{{6 \times 5 \times 4 \times 3!}} \\
   = \dfrac{1}{{120}} \\
$
So, the required probability $ = \dfrac{1}{{120}}$
Hence the option (D) is correct.
Note: We have fixed positions first, second, third for team A, B and C so that we could find the possibility of the rest of the reams and to find required probability we divided it by the total possibilities.