What does "random sampling" (usually) mean? I have two definitions.
(1) A sampling procedure is random if all individuals have the same chance of being sampled.
(2) A sampling procedure is random if all samples have the same probability of occurring.
Consider this example. I have a population of 1000 individuals, 500 men and 500 women, and I take a sample of size 100 with the constraint that 50 of the selected individuals must be men and 50 women. This would be considered random by (1) (there is nothing that makes an individual more likely to be selected), but not by (2) (some combinations, like 2 men and 98 women, aren't possible).
EDIT:
GENIVI-LEARNER has pointed out that my example doesn't fit either definition, because chances of a men being selected decrease after a men is selected. This is true only if sampling is seen as a repeated selection of one individual, and we require that all these events obey the same probability model. But if this is true, then sampling without replacement can never be random. That is, if I take out the selected item from the population after its selection, I won't have random sampling. However I have seen the adjective random applied also to this situation. So I am more inclined to think that this isn't the case.
 A: Your sample is a random sample of the men but not a random sample of the population.  When you talk of a random sample you need to specify what universe you are sampling from.  Once you restrict the universe to the men there is no difference between your two definitions.
A: First of all both of your definitions (a) and (b) are equivalent. Its just wording are changed. Having "same chance" also means same "probability of being picked". If you have both men and women in a sample then you have "defined" your population that way, so your population doesn't differentiate between men and women and you can call them "people". You can pick up the sample at random, take a note and put it back and repeat, then it satisfies both (a) and (b). Say you have 1000 samples (men and women) and say 500 of them are men then by this process you will have 1/1000 of (probability/chance) of selecting any "individual".
However, you also want to add a condition that you want to sample a total of 100 people, and if you have filled a quota of 50 men or 50 women then you wont be sampling that "class" again, so say in the process of random sampling trial you reached a point where this quota has been fullfilled for either men or women, then the probability/chance of that class changes from 500/1000 to 0. (remember the probability of individuals irrespective of class (men or women) is 1/1000 and probability of class (men or women) is 500/1000, so once you have fullfilled the quota you are dropping the probability of "class" from 500/1000 to 0/1000, in other words if you pick the next sample and if its a man and you have already sampled 50 men and fullfilled the quota, then you will ignore that sample and re-sample, (this doesnt sound random, does it, cause you now are cherry picking). So in brief initially the process was random but once a quota of any class has been full-filled then its targeted selection because you will be not be sampling the population (men or women) fairly. However, if you actually exclude all men from the sampling so now you have redefined your "population" to be "women" then the size of your population now is 500 and the probability or chance for picking up individual is 1/500 and you can proceed with random sampling as before.
