JEE (Joint Entrance Examination) consists of two phases: Screening, consisting of objective questions and Mains, consisting of subjective questions. Screening plays a major role in determining your entry - not only you need to be accurate, you also need to be fast.
In this article, let us see how can we avoid actual calculations and reduce time to arrive at the correct solution.
More often than not, you will not be required to completely solve the problem. You would be able to reject some answers by mere inspection. For example, 234*567 would be greater than 10,000. Now, 25*55 = 1375 (quick, mental calculation by Vedic Maths trick). So, the answer must be somewhere close to 137500 but less than it. How? Guess. Failed? Mail me!
Ok... that may be an artificial example. Let's take a real example - from 2004 screening.
Three distinct numbers are selected from first 100 natural numbers. The probability that all the three numbers are divisible by 2 and 3 is: (a) 4/25 (b) 4/35 (c) 4/55 (d) 4/1155
According to requirement, the numbers must be divisible by 6. There are 16 numbers divisible 6. So, the probability is:
16C3 / 100C3
16*15*14 16*15
----------- = ----------- (98/14 = 7)
100*99*98 100*99* 7
16
< ---------- (1/99 < 1/96)
100*6*7
4 4
< -------- < -------
25*6*7 100*7
The only number that is less than 4/700 is 4/1155, hence the answer.
These calculations given above are very trivial, mere approximations. You never did an actual calculation. No calculation was complex. Well, except for 98/14 = 7, if you consider it. 
