Question

How many 5 digit numbers can you create that are less than 70 000 if you can only use the digits 0,3,4,6,8 or 9.

a) Repetition of digits is permitted

b) Repetition of digits is NOT permitted

Answer 1

a) Repetition of digits is permitted 

Represent each choice with a box. The number must be less than 70000. Then the first place can be filled in 3 ways (with 3, or 4, or 6). Fill this in first, then fill the other places.

Only use 5 boxes, as our choice is restricted to 5 digits at a time.

$$3\times6\times6\times6\times6=3888$$

b) Repetition of digits is NOT permitted

Represent each choice with a box. The number must be less than 70000. Then the first place can be filled in 3 ways (with 3, or 4, or 6). Fill this in first, then fill the other places.

The second place can be filled in 5 ways.

The third place can be filled in 4 ways.

The fourth place can be filled in 3 ways.

The fifth place can be filled in 2 ways.

$$3\times5\times4\times3\times2=360$$

Or

The number must be less than 70000. Then the first place can be filled in 3 ways (with 3, or 4, or 6).

Second, third, fourth and fifth places:

The number of permutations of 5 distinct objects taken 4 at a time, where repetitions are not allowed, is given by

$${5! \over (5-4)!}=120$$

Hence the number of 5 digit numbers that are less than 70 000 is

$$3\cdot120=360$$