Question

Eight people enter a race. If there are no ties, in how many ways can the first two places come out

Answer 1

Since there is no ties:

The 1st place could be obtained in 8 ways (each one could be first)
Now that the 1st place is occupied, the remaining pretenders to te 2nd place are 7 people 

Answer 2

The number of ways the first two places can come out is 56.

Permutation and Combination

Permutation helps us to know the number of ways an object can be arranged in a particular manner. A permutation is denoted by 'P'.

The combination helps us to know the number of ways an object can be arranged without a particular manner. A combination is denoted by 'C'.

$^nP_r = \dfrac{n!}{(n-r)!},\ \ ^nC_r = \dfrac{n!}{(n-r)!\times r!}$

where,

n is the number of choices available,

r is the choices to be made.

Given to us,

Eight people enter a race, n = 8,

There are no ties, r = 2,

As there is an equal chance of everyone being first and second, but also there will be cases where the first and second positions can be exchanged between the same two people. therefore, there is a particular order in which these 8 people can win. Thus, we will use permutation.

Permutation

$^8P_2 = \dfrac{8!}{(8-2)!}=\dfrac{8!}{(6)!} = \dfrac{8\times 7\times 6!}{6!} = 8\times 7 = 56$

Hence, the number of ways the first two places can come out is 56.

Learn more about Permutation and Combination:

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