Brent@GMATPrepNow wrote:

gmattesttaker2 wrote:
How many different five-digit codes can be picked from the digits 1 through 6 if the middle digit must be odd and no two digits might be the same?

A) 420
B) 360
C) 180
D) 120
E) 60

OA: D

I was trying to solve as follows:

Middle digit could be 1 or 3 or 5.

Hence we have, 5 x 4 x 3 x 3 x 2 = 360

Can you please tell me where I am going wrong?

Thanks a lot,
Sri

Hi Sri,

Your solution looks good.
For more clarity, let's examine it step by step.

Take the task of building 5-digit numbers and break it into stages. We'll start with the most restrictive stage.

Stage 1: Select the middle digit
This digit can be 1, 3 or 5, so we can complete stage 1 in 3 ways

Stage 2: Select the 1st digit
There are now 5 digits remaining to choose from, so we can complete this stage in 5 ways

Stage 3: Select the 2nd digit
There are now 4 digits remaining to choose from, so we can complete this stage in 4 ways

Stage 4: Select the 4th digit
There are now 3 digits remaining to choose from, so we can complete this stage in 3 ways

Stage 5: Select the 5th digit
There are now 2 digits remaining to choose from, so we can complete this stage in 2 ways

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus build a 5-digit number) in (3)(5)(4)(3)(2) ways ([spoiler]= 360 ways[/spoiler])

Cheers,
Brent

Aside: For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775