Answer:
Number of outcomes = 432
Step-by-step explanation:
The outcome space will be;
S = {(x1, x2,.…… x6), xi∈{1,2,……6}∀i}
Now, S has all the ordered permutations of the numbers 1-6 which represents horses.
|S| = 6!
So, number of elements in S is 6!
Now, for the event that the number 1 is either x1, x2 or x3;
Let's call the event A;
A = {(1,x2,......x6),(x1,1,....x6),(x1,x2,1,.....x6); xi∈{2,……6},∀i}
So, there are 5! permutations of the other elements of the vector (2,3,4,5,6) and 1 can either be the first, second or third. Thus, by multiplication principle of counting;
|A| = 3 • 5!
Now,let the event where the number 2 is x2 to be B.
Thus;
B = {(x1, 2,...,x6), xi ∈ {1,3,…6}∀i}
So, there are 5! permutations of the other elements of the vector (1,3,4,5,6)
Hence, |B| = 5!
Thus, the event A ∩ B where number 1 is the first three and number 2 is second is represented as;
A ∩ B = {(1,2,...x6), (x1,2,1,...x6); xi ∈ {3,……6},∀i}
Now,the remaining numbers, 3,4,5 and 6 can be permitted in 4! ways and either 1 & 2 can be the first and second number or the third and second number.
Thus, |A ∩ B| = 2 • 4!
From set derivations, we know that,
|A ∪ B| = |A| + |B| - |A ∩ B|
So, plugging in the relevant values, we obtain ;
|A ∪ B| = (3 • 5!) + (5!) - (2 • 4!)
= (3 x 120) + 120 - (2 x 24) = 432
