Question

How many numbers less than 600 can be formed using only digits between 3 and 7?

MEDAL!!!

Answer 1

@freckles

Answer 2

If we are simply dividing 600 by 3 the answer is 200 numbers for 3 and 85 numbers for 7.

Answer 3

And honestly this seems to be some division just reworded.

Answer 4

It's actually a permutation problem

Answer 5

I see.

Answer 6

I thought 7 could not be permuted.

Answer 7

3, 4, 5, 6, 7

Number starting with 3: 1 x 5 x 5 = 25
(1 possibility for having 3 in the hundreds place, 5 possibilities for having any other number in the tens, 5 possibilities for having any other number in the ones)

Number starting with 4: 1 x 5 x 5 = 25
(1 possibility for having 4 in the hundreds place, 5 possibilities for having any other number in the tens, 5 possibilities for having any other number in the ones)

Number starting with 5: 1 x 5 x 5 = 25
(1 possibility for having 5 in the hundreds place, 5 possibilities for having any other number in the tens, 5 possibilities for having any other number in the ones)

We can't have a number starting with 6 because it has to be less than 600.

So I added all of the ways I got (25 + 25 + 25) and then I got my answer as 75.

Answer 8

That seems correct to me after double reading it.

Answer 9

But the answer key says that it's 105.

Answer 10

Oh? List the answers.

Answer 11

There aren't any choices.

Answer 12

Ah. Well I can give an example, that I hope would help. 6 numbers involved this time.
The numbers are 2,3,5,6,7,9
3 numbers out of 6 can be chosen and permuted in 6P3 = 6! / (6-3)! =
= 6 x 5 x 4 = 120 ways
120 numbers can be formed

There are 3 spaces x x x
The first space can be filled by 2 and 3 only since the number < 400.
The second space can be filled in 5 ways
The third in 4 ways
2 x 5 x 4 = 40 3-digit numbers less than 400.

Even numbers end with 2 and 6
x x x
The 3rd space can be filled in 2 ways.
The first space can be filled in 5 ways and the second in 4 ways
5 x 4 x 2 = 40

Multiples of 5 end with 5
x x x
Last space in 1 way, namely number 5.
First and second spaces can be filled in 5 and 4 ways
5 x 4 x 1 = 20 digits are multiples of 5

Answer 13

I still don't understand how they got to 105!

Answer 14

Okay. I think I got it. I used the same process written above and just added two extra steps.

Since it didn't say specifically that it had to be a three digit number, then there could be two digits and one digit numbers too.

Two digit, you can have all of the five numbers in the tens and hundreds place as it will still be less than 600. (5 x 5 = 25)

One digit, you can have all of the five numbers and it will be less than 600. (5)

75 + 25 + 5 = 105 ways

Answer 15

good thinking.... I miss that it didn't have to be three digits at first
good eye
If it was 3-digits I would have done this:
How many 3-digit numbers less than 600 can be formed using only digits between 3 and 7?


so we want numbers less than 600

so let xyz be a digit number where x is a hundreds digit
and y is the tens digit
and z is ones digit

....

we want x to be a number between 3 and 5
y can be any number between 3 and 7
and z can be any number between 3 and 7


so there are only 3 ways to choose x
and 5 ways to choose y
and 5 ways to choose z

so 3*5*5=75