Question

How many different telephone numbers are there if it is assumed that each number contains not more than seven digits (the first digit is not allowed to be $ 0{\text{ or 1}} $ in a telephone number)?

Answer 1

Hint: In order to deal with this question we will determine all the possible condition for each digit but we have to keep in mind that number should not start from $ 0{\text{ or 1}} $ , so according to it we will make the case for each digit and at last by multiplying all of them we will get the required answer.

Complete step-by-step answer:
We have to form $ 7 $ digits number by keeping in mind that first digit is not allowed to be $ 0{\text{ or 1}} $

So, 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 digit possibilities; let us explain:

Let’s say the first digit 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:

\[30,31,32 \ldots 39\]
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)
\[\begin{array}{*{20}{l}}
  {10\left( {{\text{ }}for{\text{ }}2nd{\text{ }}digit} \right)} \\
  {10\left( {{\text{ }}for{\text{ }}3rd{\text{ }}digit} \right)} \\
  {10\left( {{\text{ }}for{\text{ }}4th{\text{ }}digit} \right)} \\
  {10\left( {for{\text{ }}5th{\text{ }}digit} \right)} \\
  {10\left( {{\text{ }}for{\text{ }}6th{\text{ }}digit} \right)} \\
  {10\left( {{\text{ }}for{\text{ }}7th{\text{ }}digit} \right)}
\end{array}\]

So, we get a total of \[8 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10{\text{ }} = {\text{ }}8,000,000\] phone numbers possible.

Hence the required number of possible telephonic numbers are \[8,000,000\].

Note: A mobile number contains \[10\] digits with the help of permutations we can find the different numbers that depend on the factorial of the possibilities that shows the different way of replacing numbers.