Question

How many different seven-digit phone numbers do not contain the digits 6 or 7?

Answer 1

2,097,152 possibilities

Explanation:

out of the 10 possible digits, two of them are not available (the 6 or 7), this leaves us with 8 usable digits.

Using a calculator, you multiply the 8 times each of the seven digits in a phone number. 8 to the power of 7.

8^7=2,097,152

Answer 2

Explanation:

Think of it this way -

There are 8 different possibilities for the first number: the numbers 2–9

For the second number there are ten possibilities: 0–9

The trick here is that there are 10 possibilities for each one of the first number possibilities; let me explain:

Let’s say the first number is 2.

Then there are ten different possibilities for the first two digits:

20,21,22,23,24,25,26,27,28,29

So, that’s a total of 10, now let’s assume that the first digit is 3:

Now there are ten more possibilities:

30,31,32…39

So our total is now 20.

We keep repeating this for each of the first digit possibilities and get 80 possibilities, for the first two digits. Now, for each of these possibilities, there are another ten possibilities each for the third digit, making a total of 800 possibilities. Now we can easily see that this process will be repeated for the next 4 digits, and we get for the final answer:

8(possibilities for the first digit)*10(…2nd digit)*10(3rd)*10(4th)*10(5th)*10(6th)*10(7th

So, we get a total of 8*10*10*10*10*10*10 = 8,000,000 phone numbers possible

Interestingly, this is why phone numbers have area codes, because there are far more than 8 million people who need phone numbers, so three more digits added to the number to create a total of 8,000,000*10*10*10 = 8,000,000,000 phone numbers! This will last for quite a while, considering that not everyone in the world’s 7.6 billion people has a phone, or ever will have a phone. However, when we do run out, we can just add another digit to the front to get 80 billion possible numbers!