There are three secretaries who work for four departments. If each of the four departments have one report to be typed out, and the reports are randomly assigned to a secretary, what is the probability that all three secretary are assigned at least one report?
Choose the correct choice from the following choices.
A) 8/9	B) 64/81	C) 12/27	D) 16/81	E) 5/9
"We havé total possibilities N =3^4"   What are these total possibilities? They are the ways in which the 4 reports can be distributed among 3 secretaries so that all reports may go to one secretary, the reports may be distributed among 2 secretaries or they may be distributed among all 3. So such cases are included (4, 0, 0), (0, 4, 0), (2, 2, 0), (1, 3, 0), (1, 1, 2), etc   In how many ways can you give 4 reports to only 1 secretary? You choose the secretary who will get the reports in 3C1 = 3 ways   In how many ways can you give 4 reports to only 2 secretaries? Choose the 2 secretaries in 3C2 ways. Each secretary must get at least one report so you can distribute them in 2 ways:   '1 and 3' - Choose one report in 4C1 ways and give it to one secretary in 2C1 ways. The other secretary gets the other 3 reports.   or   '2 and 2' - Choose 2 reports in 4C2 ways and give to the first secretary. The other secretary gets the other two reports.   Total number of ways is 3C2 * (4C1*2C1 + 4C2) = 42 ways   Note that in your method 2^4 is incorrect. It includes the ways in which all 4 reports go to one secretary.   So this means that in  3 + 42= 
<gadget id="calculator"> 3 + 42</gadget>
<output>45</output>
45 ways, at least one secretary gets no report. In the rest of  81 - 45= 
<gadget id="calculator"> 81 - 45</gadget>
<output>36</output>
36 ways, each secretary gets at least one report.   Required Probability= 36/81 = 12/27   ANS:12/27
<result>C</result>
There are three secretaries who work for four departments. If each of the four departments have one report to be typed out, and the reports are randomly assigned to a secretary, what is the probability that all three secretary are assigned at least one report?