Here are my favorite shortcuts:
1. Number of Intergers between A, B:
(B-A)+1 : Sounds simple but then very useful for tough problems.
Number of integers between 100 and 1000 that are divisible by 2 or Number of even integers bet' 100 and 1000:
starting point, A = 102, End point, B = 998
number of even integers = (998-102)/2 + 1
2. Fill in the blanks method to find the number of combinations:
Eg: Find the number of odd 3 digit integers, which do not contain the number 5 in them and no digit is repeated:
Trick: draw three blank spaces (representing the 3 digits) like so _ _ _
(i) Since we need odd integers, the number of possible integers that be be used to fill the rightmost blank is
4 (1,3,7,9 remember no 5)
(ii) Second blank space could be filled by
8 numbers (out of 10 possible digits take out 5 and take out one which was allocated for the third blank - no repeats)
(iii) Third blank can be filled by
6 digits (no zero, no five, and take 2 digits that were occupied by previous blanks)
(iv) So the total possibilities are = 4*8*6 = 192 !!!
It might look long but thats coz i'm not good at typing in my explanation
3. To find the rate of growth:
A = P (1+r)^t
A= Final value
p=Initial Value
r=rate of growth per period
t=total time to grow from P to A
This cool formula can save a lot of time in Interest problems, those wicked bacterial growth problem etc.