[ Classical combinatorics (other).]

Given a six-digit phone number. From how many seven-digit numbers can it be obtained by erasing one digit

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How many ways are there to insert one more digit into a phone number?

## Solution

Let $A B C D E F$ be a six-digit phone number. Let's count how many seven-digit numbers can be obtained from it by adding one digit. Before the number, you can prepend any of the 10 digits, but the number obtained by prepending the digit $A$ can also be obtained by inserting the digit $A$ between $A$ and $B$. We will account for this case later. So, for now, we have 9 numbers. Similarly, we can get 9 numbers by inserting a digit between $A$ and $B$ (the number obtained by inserting the digit $B$ there will be obtained in the next step), between $B$ and $C$, and so on. Finally, 10 numbers can be obtained by appending a digit to the end of the original number. Thus, we get $6 \cdot 9 + 10 = 64$ numbers.

## Answer

64 numbers