Dear Abhinav Khanna,
That's a great question.
The reason we
don't have to subtract 5 is because of the way permutations are designed. When we take a permutation of k elements from a pool of n, implicit in that process is that (a) the objects distinct from one another, and (b) we are selecting them
without replacement. If Q is an element of the pool of n, then Q will either be selected or not selected, but if it is selected, there is no possibility of its being selected twice.
If you think about it, that's where the whole factorial idea comes from for permutations. For example, consider 4P4, the permutations of a set of four distinct objects --- 4P4 = 4! = 4*3*2*1 = 24. The reason is: for the first position, you have four choices; then for the second position, you have three choices --- the three remaining after the first choice; for the third position, you have two choices, the two remaining after the first two positions are chosen; then, only one choice remains for the last position. That is all assuming --- no object is going to be chosen more than once: when an item is chosen, it is "used up", no longer available. This means: no repeats.
If we could repeat as much as we wanted, then there would be four choices available for each of the four places, and the total number would be 4^4 = 256. That's something quite different from a permutation.
If you want to allow for the possibility of repeat items --- for example, digits where double digits are allowed, then permutations are
not the appropriate tool to use. You would either have to (a) calculate the number of permutations, then add in the doubles, or (b) just stick with the
Fundamental Counting Principle (which is always a port in a storm).
Here's a free video about the FCP.
https://gmat.magoosh.com/lessons/335-fun ... -principle
Again, that was a very perceptive question. Does all that I said make sense to you? Please do not hesitate to ask if you have any further questions.
Mike
