# differentiating “n” question

i would like to know the difference here (this is no longer a homework question but rather concept differentiation)

we know that probability = s/n (where s = success and n = total outcomes)

in this question:

    For a certain style of new automobile, the colors
blue, white, black, and green are in equal demand.
Three  successive  orders  are  placed  for  automo-
biles of this style. Find the following probabilities
a  One  blue,  one  white,  and  one  green  are
ordered.
b  Two blues are ordered.
c  At least one black is ordered.
d  Exactly  two  of  the  orders  are  for  the  same
color.


the "n" is simply 4 x 4 x 4 = "64" in each letter

while in this type of question

An  assembly  operation  for  a  computer  circuit
board  consists  of  four  operations  that  can  be
performed in any order.
a  In how many ways can the assembly opera-
tion be performed?
b  One  of  the  operations  involves  soldering
wire to a microchip. If all possible assembly
orderings  are  equally  likely,  what  is  the
probability that the soldering comes first or
second?


the "n" in letter b is "24" (which is simply 4! or 4 x 3 x 2 x 1)

whats the difference of n between the 2 questions. im having difficulty to understand n here.

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What is the word concept referring to, here? – Did Jan 22 '12 at 15:48
its referring to differentiating the "n" midst various situations – WantIt Jan 23 '12 at 1:33
Then use difference or whatever, but not concept. – Did Jan 23 '12 at 1:37
ok edited. tnx. – WantIt Jan 23 '12 at 5:51

Firstly, some assumptions are being made. We should be explicit about them, so that we understand what's going on. We are assuming every outcome to be equally likely. This is a very special assumption. It's what lets us say, since outcome X can happen in 10 different ways, and there are 20 total possibilities, we deduce that outcome X has probability $1/2$.

Other than that, the only key idea is that the first question allows replacement. So we might have Blue, Blue, Blue. In the second, we cannot have Solder, Solder, Solder, Solder. In elementary questions, there are only a few different aspects that can change. The 4 most important are: the number of objects, how many are to be chosen, whether they are chosen with replacement (i.e. whether they can or cannot repeat - this is what's going on here), or whether different objects can be distinguished.

If you can identify these 4 aspects fully, you likely have gotten all the information you need to go to some form of formulae or other direct computation.

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In the first question, you have 4 choices for the color of the first car, and for each such choice you have four choices for the color of the second car, and similarly for the third. So you have $4\times 4\times 4$ possibilities.

In the second example, after you have made one of the four choices for the first operation, you are not allowed to use that operation as the second operation again. So you have only 3 choices for the second operation. Similiarly, you are not allowed to use one of the first two choices for the third operation, so only two remain. After this choice is made, only one operation remains. So you have $4\times 3\times 2\times 1$ possibilities.

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thx! ur explanation helps – WantIt Jan 22 '12 at 15:07