Resolve into partial factors : \(\frac { { x }^{ 2 }+x+1 }{ { x }^{ 2 }+2x+1 } \)

In how many ways can the following prizes be given away to a class of 30 students, first and second in mathematics, first and second in physics, first in chemistry and first in English?

How many numbers are there between 100 and 1000 such that atleast one of their digits is 7?

If 22P_r+1:20P_r+2=11: 52, find r.

How many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4, 2, 3?