Counting problem in probability How many 7-digit phone numbers are possible assuming that the first 3 digits can't start with 911?
I should add, I just started studying probability and watching lectures on youtube. The first part of this question is to assume the first digit can't be a 0 or 1 which I solved; 8 possibilities for the first number( ignoring 0 and 1), 10 possibilities for the rest. I know the answer to the problem I posted (8 x 10^6-10^4) but I don't understand the answer and I'd like an explanation which is why I posted it here. You don't have to respond to this post if you don't want to, there's no need to be condescending. It may be elementary to some of you but I don't understand it.
 A: The multiplication principle, paraphrased, states that if you want to count the number of outcomes a scenario has... if you can describe each and every outcome uniquely via a sequence of steps such that the number of available options for a particular step is consistent and does not rely on previously made choices at earlier steps (though the list of options may vary)... then the total number of outcomes is the product of the number of options for the steps.
In terms of set theory, this is commonly stated as:
$$|A\times B| = |A|\times |B|$$
Next, the addition principle... if you want to count the number of outcomes a scenario has and you can partition the set of outcomes into two smaller disjoint exhaustive sets, you can count those smaller sets and add their respective results to get the larger set.
In terms of set theory, this is commonly stated as:
$$|A\sqcup B| = |A|+ |B|$$
(Where $\sqcup$ is the union symbol but carries with it the connotation that it is understood that $A$ must be disjoint from $B$ to be used)

 Note, the more general identity is $|A\cup B|=|A|+|B|-|A\cap B|$ where we drop the requirement that $A$ and $B$ must be disjoint.  The idea here is that we might have double-counted those outcomes which were in the overlap of $A$ and $B$ and the overall count needs to then be corrected to account for this.

We can rearrange this last one by subtracting to the other side to get the related identity that to count the number of "good" outcomes, we can count the overall number of outcomes and subtract the number of "bad" outcomes, that $|A\sqcup B|-|B|=|A|$

So... using these... we count the number of overall outcomes for the number of 7-digit phonenumbers if we don't care about it starting with 911.
Given the solution, it is assumed that a phone number can not start with $0$ or with $1$.

*

*Pick the first number.  There are $8$ choices

*Pick the second number.  There are $10$ choices

*Pick the third number.  There are $10$ choices

*Pick the fourth number.  There are $10$ choices

*$\vdots$
Applying multiplication principle, there are then $8\times 10\times 10\times \dots \times 10 = 8\times 10^6$ different 7-digit phone numbers.
Now, to count the "bad" phone numbers from those which were those who started with 911.

*

*The first three numbers are 9,1,1 respectively.  There is only the one choice available for these for the number to have started with 9,1,1.

*Pick the fourth number.  There are $10$ choices

*Pick the fifth number.  There are $10$ choices

*$\vdots$
Multiplying these number of options together, that gives a total of $1\times 10\times 10\times 10\times 10 = 10^4$ "bad" phone numbers that we had accidentally included in our count earlier.
Taking the difference gives the final result as being:
$$8\times 10^6 - 10^4$$
