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A footrace takes place among four runners. If ties are allowed (even all four
runners finishing at the same time), how many ways are there for the race to
finish?

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Here we have to consider the different cases based on number of ties :

a) No tie :

In that case , 4 individuals will complete race in different times..So 4! ways possible..Hence number of ways in this case = 24

b) One tie involving 2 persons : 

Here we have to choose 2 persons first between whom tie will occur which can be done in C(4,2) =  6 ways..Having done this , considering the two people having tie as one unit , we have 3 units now instead of 4 hence they can be arranged in 3! = 6 ways..Hence total no of ways in this case  =  36 ways..

c) One tie involving 3 persons :

Here we choose 3 persons first which can be done in C(4,3) = 4 ways..So here we have 2 units and hence can be arranged in 2! = 2 ways..Hence in total , no of ways in this case    =   4 * 2  =  8 ways.

d) Two ties involving 2 persons in each tie :

Here we select 2 persons first which is done in C(4,2)  =  6 ways..Remaining 2 persons will be in another group and will tie automatically..So here we have 2 pairs and arrangement among pair is possible..Hence no of ways  =  6 * 2  = 12 ways

e) One tie involving all 4 persons :

This can be done in only one way..

Hence number of ways for race to finish   =   24 + 36 + 8 + 12 + 1

                                                             =   81 ways

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