1. ## Counting Question

Alright guys, here's the question.

How many five digit numbers with no beginning zeros (10000 - 99999) meet at least one of the following requirements?

- Starts with a 3
- Middle digit is 5
- Last digit is 7

Starts w/ 3:The first digit only has one way to be made, the remaining 4 each have 10 possibilities. Therefore 1 X 10^4
Middle Dig 5::Because the first digit can't be 0, we have 9 ways to make the first digit, 10 ways for 2,4,5th digit, and one way for 3rd. Therefore 1 X 9 X 10^3
Last Dig 7: Again we see first digit has 9 way, digits 2-4 each have 10 ways, and final digit has 1 way. Therefore 1 X 9 X 10^3.

With this in mind, the number of numbers that meet at least one of the following is simple the number from each set added up?

1 X 10^4 X 9^2 X 10^3 X 10^3

Does this look correct?

Thanks!

2. ## Re: Counting Question

first off you'd sum the 3 not multiply them

secondly you have to worry about numbers that satisfy 2 or more of the criteria and only count them once

3. ## Re: Counting Question

Meant to sum, hence the "added up" above the multiplication, lol.

4. ## Re: Counting Question

Originally Posted by PodoTheGreat
How many five digit numbers with no beginning zeros (10000 - 99999) meet at least one of the following requirements?
- Starts with a 3
- Middle digit is 5
- Last digit is 7.
Use the following notation $\mathcal{T}$ is the set of those numbers starting with three; $\mathcal{F}$ is the set of those numbers having middle digit five; $\mathcal{S}$ is the set of those numbers having last digit seven.
The notation $\|\mathcal{A}\|$ is the number of terms in the set $\mathcal{A}$.

This question is asking for $\|\mathcal{T\cup F\cup S}\|$ which means that you must know the inclusion/exclusion rules.

$\|\mathcal{T\cup F\cup S}\|=\|\mathcal{T}\|+\|\mathcal{ F}\|+\|\mathcal{ S}\|-\|\mathcal{T\cap F}\|-\|\mathcal{T\cap S}\|-\|\mathcal{ F\cap S}\|+\|\mathcal{T\cap F\cap S}\|$

Here is a start $\|\mathcal{F}\|=(9)(10)^3$ & $\|\mathcal{T\cap S}\|=(10)^3$

5. ## Re: Counting Question

Originally Posted by romsek
first off you'd sum the 3 not multiply them
secondly you have to worry about numbers that satisfy 2 or more of the criteria and only count them once

6. ## Re: Counting Question

Originally Posted by romsek
secondly you have to worry about numbers that satisfy 2 or more of the criteria and only count them once
So, the number of substrings that start w/ 3, mid dig 5, and last dig 7 = 1(10)1(10)1 = 10^2.
Since that covers all strings that start w/ 3 and end with 7, start w/ 3 and middle digit 5, as well as middle digit 5 and end w/ 7, we simply subtract 10^2 from the final answer?
Would that account for it?

7. ## Re: Counting Question

Originally Posted by Plato
Use the following notation $\mathcal{T}$ is the set of those numbers starting with three; $\mathcal{F}$ is the set of those numbers having middle digit five; $\mathcal{S}$ is the set of those numbers having last digit seven.
The notation $\|\mathcal{A}\|$ is the number of terms in the set $\mathcal{A}$.

This question is asking for $\|\mathcal{T\cup F\cup S}\|$ which means that you must know the inclusion/exclusion rules.

$\|\mathcal{T\cup F\cup S}\|=\|\mathcal{T}\|+\|\mathcal{ F}\|+\|\mathcal{ S}\|-\|\mathcal{T\cap F}\|-\|\mathcal{T\cap S}\|-\|\mathcal{ F\cap S}\|+\|\mathcal{T\cap F\cap S}\|$

Here is a start $\|\mathcal{F}\|=(9)(10)^3$ & $\|\mathcal{T\cap S}\|=(10)^3$
I can't quite follow what you're saying because of the formatting. Is that latex?

8. ## Re: Counting Question

Originally Posted by PodoTheGreat
I can't quite follow what you're saying because of the formatting. Is that latex?
It has nothing whatsoever with LaTeX.
Do you know anything about inclusion/exclusion?
If not, your trying to answer this question is a waste of time.

9. ## Re: Counting Question

100 meet the "3-5-7":

1: 30507
2: 30517
...
99: 39587
100: 39597

10. ## Re: Counting Question

Originally Posted by PodoTheGreat
You're code didn't post correct and looked like a bunch of gibberish, not equations.
What web-browser do you use. If you are not using a phone or a pad, but are using one of the most popular browsers you have reading the code.
We have tested on many platforms.

Here is a work-around. Click the reply with quote. Then remove all code tags: remove all [ tex][/tex][ /tex][/tex] tags & all \$'s.