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