Question 1159882
.
7 men and 3 women are ranked according to their scores on an exam. Assume that no two scores are alike, 
and that all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a man 
(so X=1 indicates that a man achieved the highest score on the exam). Find each of the following:
(a) P(X=1) =
(b) P(X=2) =
(c) P(X=3) =
(d) P(X=7) =
~~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
(a)  X=1  means that one of the 7 men is in the first position,
          while the rest 6 men and 3 women are distributed in positions from 2 to 10 in arbitrary ways.

     The number of such possible outcomes is {{{C[7]^1*9!}}} = 7*9!,
     and to find the probability  P(X=1),  we should relate 7*9!  to  10!.


     It gives  P(X=1) = {{{(7*9!)/10!}}} = {{{7/10}}} = 0.7.     


     It is the  <U>ANSWER to (a)</U>.



(b)  X=2 means that one of the 3 women is in the first position and one of the 7 men is in the second position,
         while the rest 3-1 = 2 women and 7-1 = 6 men are distributed in positions from 3 to 10 in arbitrary ways.

     The number of such possible outcomes is {{{C[3]^1*C[7]^1*(2+6)!}}} = 3*7*8!,
     and to find the probability  P(X=2),  we should relate  3*7*8!  to  10!.


     It gives  P(X=2) = {{{(3*7*8!)/10!}}}= {{{(3*7)/(9*10)}}} = {{{7/(3*10)}}} = {{{7/30}}}.


     It is the <U>ANSWER to (b)</U>.



(c)  X=3 means that two of the 3 women are in the first and in the second positions and one of the 7 men 
         is in the third position,
         while the rest 3-2 = 1 women and 7-1 = 6 men are distributed in positions from 3 to 10 in arbitrary ways.

     The number of such possible outcomes is  {{{C[3]^2*C[7]^1*(6+1)!}}} = {{{((3*2)/2)*7*7!}}} = 3*7*7!,
     and to find the probability  P(X=3),  we should relate  3*7*7!  to  10!.


     It gives  P(X=3) = {{{(3*7*7!)/10!}}}= {{{(3*7)/(8*9*10)}}} = {{{7/(8*3*10)}}} = {{{7/240}}}.


     It is the <U>ANSWER to (c)</U>.



(d)  X=7  means that some man is in the 7th positions and there no men in positions from 1 to 6.

          But this configuration is not possible (which is obvious), so

              P(X=7) = 0.


     It is the <U>ANSWER to (d)</U>.
</pre>

Thus, all the questions are answered, and the problem is solved completely.