### Lesser Digits

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

### Mini-max

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers and so on?

### The Codabar Check

This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.

# Six Times Five

##### Age 11 to 14 Challenge Level:

Firstly consider the number of six digit numbers - this is 900,000.

$\frac19$ of all six digit numbers start with a 5. So 100,000 six digit numbers are of the form 5******

$\frac1{10}$ of the remaining numbers have a 5 in the ten-thousands column, so we need to subtract 80,000 from 800,000 leaving 720,000.

$\frac1{10}$ of the remaining numbers have a 5 in the thousands column, so we need to subtract 72,000 from 720,000, leaving 648,000.

$\frac1{10}$ of the remaining numbers have a 5 in the hundreds column, so we need to subtract 64,800 from 648,000, leaving 583,200.

$\frac1{10}$ of the remaining numbers have a 5 in the tens column, so we need to subtract 58,320 from 583,200 leaving 524,880.

$\frac1{10}$ of the remaining numbers have a 5 in the units column, so we need to subtract 52,488 from 524,880, leaving 472,392.

A slightly quicker method would be to multiply by 0.9 instead of subtracting $\frac1{10}$ in each of the above steps.

Here is a different solution, from Junwei of BHASVIC

Let the six digits number is abcdef, which a, b, c, d ,e, f represent a digit respectively.

For a, neither 0 nor 5 could place in it, thus, 8 digits are available here (1,2,3,4,6,7,8,9)

For b, c, d, e and f, they can't contain 5, hence, 9 digits are available for them (0,1,2,3,4,6,7,8,9)

Therefore, the no. of six digits number which does not contain any 5 is

8 * 9 * 9 * 9 * 9 *9 =472392 .