Permutations and Combinations

Let the number of students in the class be n.
Let a be the age of youngest student
∴                              a = 8 years
Since the age of the students form an A.P. with common difference 4 months or  years.
∴                              years
Since the sum of the ages of all the students in the class is 168 years.
∴                                      
                          
                                 

                  
                            
                     n = 16
∵   Number of students cannot be negative. Hence, number of students in the class is 16.

Money deposited = Rs 10,000, Rate of interest = 5%
Interest for one year = 
Let a be the amount deposited and d be the interest every year.
∴                           a = 10000,  d = 500
Amount in the 15th year = 
Let   denote the total amount after 20 years
                       
                              = 10 [29500] =  Rs. 29500

Let a mm be the distance travelled in the first second. Since the insect covers half the distance covered in the previous second in the succeeding second.
∴   Distances covered in various seconds are 
Let r be the common ratio
∴                                    
Let 3 mm distance be covered in n seconds
∴                  
             
which is impossible for any value of n.
Hence, the insect would never reach a point 3 mm  away from its starting point.

Let the number of sides of the polygon be n. The interior angles from an A.P. Let a be the smallest angle and d be the common difference.
∴                    
Since the sum of all the interior angles of a polygon with n sides is 
∴            
                 
              
                                     
                               
                                                             n = 9,  16
But if n = 16, then  which is impossible.

∵  No angles of polygon exceeds 
Hence,                         n = 9