Common Information
There are $5$ freshmen, $8$ sophomores, and $7$ juniors in a chess club. Find the number of orders in which the $6$ students from this club can win the first six prizes:
Q6. 
Common Information Question: 1/4 If all students attend the competition and the winners are exactly 3 freshmen. 
A.  $3! × {^5C_3} × {^{15}C_3}$ 

B.  $3! × {^5P_3} × {^{15}P_3}$ 

C.  $6! × {^5C_3} × {^{15}C_3}$ 

D.  $6! × {^5P_3} × {^{15}P_3}$ 
Q7. 
Common Information Question: 2/4 If all students attend the competition and the winners are exactly 3 freshmen and 3 sophomores. 
A.  $3! × {^5C_3} × {^15C_3}$ 

B.  $3! × {^5C_3} × {^8C_3}$ 

C.  $6! × {^5P_3} × {^15P_3}$ 

D.  $6! × {^5C_3} × {^8C_3}$ 
Q8. 
Common Information Question: 3/4 If all students attend the competition and the winners are an equal number of freshmen, sophomores, and juniors. 
A.  $6! × {^5C_2} × {^8C_2} × {^7C_2}$ 

B.  $3! × {^5C_2} × {^8C_2} × {^7C_2}$ 

C.  $6! × {^5P_2} × {^8P_2} × {^7P_2}$ 

D.  $3! × {^5P_2} × {^8P_2} × {^7P_2}$ 
Q9. 
Common Information Question: 4/4 If all students attend the competition and the winners are all members of the same class. 
A.  21,200 

B.  23,200 

C.  25,200 

D.  27,200 
Q10. 
Find the number of ways in which $8064$ can be resolved as the product of two factors? 
A.  22 

B.  24 

C.  21 

D.  20 