# How many positive numbers can be made from the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 so that they are smaller than 800? The digits can repeat.

Nov 4, 2017

$657$ $\text{numbers}$.

#### Explanation:

These are numbers less than $800$, so they are at most $3$ digit numbers.

So, make three slots:

## _ _ _

The first slot can be filled with only $1 , 2 , 3 , 4 , 5 , 6 \mathmr{and} 7$ (7 "digits") because the number is less than $800$. So, it can be filled in $7$ ways.

Since repetition is allowed, the second and third slot can be filled in $9$ ways each.

Now, multiply the number of ways:

$\rightarrow 7 \times 9 \times 9$

color(green)(rArr567

But wait, if we think about this, we notice that we can also form two-digit numbers and one-digit numbers less than $800.$

We can form $9$ one digit numbers.

Now let's solve for the number of two-digit numbers. So put two slots:

## _ _

We can write $9$ digits in each of the slots:

So there a total of $9 \times 9 = 81$ ways.

$\rightarrow 567 + 9 + 81$
color(green)(rArr657