Question 1154280
<pre>The usable characters are these: 26 capital letters, 26 lower case letters and
10 digits.  That's the 62 characters below.

ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789

We choose the first character as any of the 52 letters.
The other characters can be any of the 62.

There are 52 ways to choose a 1-character file name.
There are 52∙62 ways to choose a 2-character file name.
There are 52∙62<sup>2</sup> ways to choose a 3-character file name.
There are 52∙62<sup>3</sup> ways to choose a 4-character file name.
There are 52∙62<sup>4</sup> ways to choose a 5-character file name.
There are 52∙62<sup>5</sup> ways to choose a 6-character file name.
There are 52∙62<sup>6</sup> ways to choose a 7-character file name.
There are 52∙62<sup>7</sup> ways to choose a 8-character file name.
There are 52∙62<sup>8</sup> ways to choose a 9-character file name.
There are 52∙62<sup>9</sup> ways to choose a 10-character file name.
There are 52∙62<sup>10</sup> ways to choose a 11-character file name.
There are 52∙62<sup>11</sup> ways to choose a 12-character file name.
There are 52∙62<sup>12</sup> ways to choose a 13-character file name.

That's 52+52∙62<sup>2</sup>+52∙62<sup>3</sup>+52∙62<sup>4</sup>+52∙62<sup>5</sup>+52∙62<sup>6</sup>+52∙62<sup>7</sup>+52∙62<sup>8</sup>+52∙62<sup>9</sup>+52∙62<sup>10</sup>+52∙62<sup>11</sup>+52∙62<sup>12</sup> ways

52(1+62<sup>2</sup>+62<sup>3</sup>+62<sup>4</sup>+62<sup>5</sup>+62<sup>6</sup>+62<sup>7</sup>+62<sup>8</sup>+62<sup>9</sup>+62<sup>10</sup>+62<sup>11</sup>+62<sup>12</sup>+62<sup>13</sup> ways


That's 52 times a geometric series with a<sup>1</sup> = 1, n=13, r=62.

{{{S[n]=a[1](r^n-1)/(r-1)}}}

{{{S[13]=1(62^13-1)/(62-1)}}} 

{{{52*3279156381453603096811}}}

170516131835587361034172

That's the answer. [A 22-digit number!!!]

Edwin</pre>