Consecutive integers problem

chaitanyareddy wrote:How many 5-digit positive integers exist where no two consecutive digits are the
same?
A.) 9*9*8*7*6
B.) 9*9*8*8*8
C.) 9^5
D.) 9*8^4
E.) 10*9^4

Here's a step by step solution using the Fundamental Counting Principle (aka "slot method")

Take the task of building a 5-digit number and break it into stages.

Stage 1: Select the 1st digit (the ten thousands digit)
The first digit can be 1,2,3,4,5,6,7,8 or 9 (can't have 0 as first digit, otherwise it's not a 5-digit number)
So, we can accomplish stage 1 in 9 ways.

Stage 2: Select the 2nd digit (the thousands digit)
Once the 1st digit has been selected, we cannot select it for the 2nd digit
So, we can accomplish stage 2 in 9 ways.

Stage 3: Select the 3rd digit (the hundreds digit)
Once the 2nd digit has been selected, we cannot select it for the 3rd digit
So, we can accomplish stage 3 in 9 ways.

Stage 4: Select the 4th digit (the tens digit)
Once the 3rd digit has been selected, we cannot select it for the 4th digit
So, we can accomplish stage 4 in 9 ways.

Stage 5: Select the 5th digit (the units digit)
Once the 4th digit has been selected, we cannot select it for the 5th digit
So, we can accomplish stage 5 in 9 ways.

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus create a 5-digit number) in (9)(9)(9)(9)(9) ways ([spoiler]= 9^5 ways = C[/spoiler])

Cheers,
Brent

Aside: For more information about the FCP, we have a free video on the subject: https://www.gmatprepnow.com/module/gmat-counting?id=775

As I showed in my earlier solution (2 posts above), we can solve the question by applying the Fundamental Counting Principle (FCP).

NOTE: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html
- https://www.beatthegmat.com/mouse-pellets-t274303.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/ps-counting-t273659.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/please-solve ... 71499.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/laniera-s-co ... 15764.html

Cheers,
Brent

mendezsk wrote:Hello All,

Please be of guidance. I am actually receiving D (9*8^4) as my answer.

Logic:

The question states how many 5 digit positives integers exist where no 2 consecutive integers are the same?

the 1st integer: 1-9
2nd integer: 8 possibilities (lets say the 1st integer was 2... the 2nd integer cannot be 2, therefore it can be 1,3,4,5,6,7,8, or 9 = 8 possibilities. Why are we including 0-9 when 0 is not a positive integer)?
3rd integer: 8 poss
4th integer: 8 poss
5th integer: 8 poss

9*8^4
That's what I am getting. So please help

You're correct when you say that 0 is not a positive INTEGER.
BUT, it's okay to include 0 as one of the DIGITS in a positive INTEGER.

For example, 50532 is a positive integer that contains the non-positive DIGIT 0.

Cheers,
Brent