Doubt in multinomial distribution Week-5 Lecture-1

[Himanshu]

I am very confused in the multinomial distributiin and have the following doubts-

i) In the lecture, sir has defined Xj = No. of trials that result in the jth outcome. We have to find P(X1=x1, X2=x2,. ......, Xk=xk) where x1+x2+ .... +xk = n.  

So, here X1, ... , Xk are all random variables ( Xi is function from S to Ti) because sir has used notation X1=x1 meaning X1^{-1}(x1). I have a doubt why we have to define k random variables. We can define one random variable X such that each outcome in S is mapped to number of trials that result in jth outcome. Then our question of joint distribution which we have to find in the example looks like P(X=T) where T = range(X) = {x1,...,xk} instead of P(X1=x1, X2=x2,. ......, Xk=xk) ? I am not able to understand like how the k random variables Xi actually meant?   

ii) Also I have not clearly understood the set B(x1,...,xk).  

I get very confused in this section. Please clarify the doubt.