# How many 7-digit phone numbers can be formed in the first digit cannot be 0, and any digit can be repeated?

Dec 13, 2016

This is an example of using an advanced counting scheme where we find the answer to be $9$ million with out having to write them all out!!. Read on...

#### Explanation:

We consider that each digit is selected independently of the others. This is a consequence of the statement that says digits can be repeated.

The first digit can be any of $1$ through $9$ and can be selected in nine different ways. Each remaining digit can then be selected in $10$ ways, as zero can be used.

Since each digit choice is independent, we just multiply all these choices together:

$N = 9 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = \text{9 million possibilities}$

(Shows you why area codes have to come into play.)