math letter counting PS

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Castor.kim: Junior | Next Rank: 30 Posts; Posts: 22; Joined: Thu Aug 30, 2012 10:23 pm

math letter counting PS

by Castor.kim » Tue Nov 13, 2012 1:19 am

It's a problem from GMAT institute.

A code consists of three of the 26 letters. The first and third letters are different consonants and the second letter is a vowel. How many such codes are possible?

it's PS problem, but not an official, so it has no option

the answer is 2100

but I don't know how to get this answer..
please someone help me..

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Source: Beat The GMAT — Problem Solving | Tue Nov 13, 2012 1:19 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Tue Nov 13, 2012 6:18 am

Castor.kim wrote: A code consists of three of the 26 letters. The first and third letters are different consonants and the second letter is a vowel. How many such codes are possible?

Take the task of creating the code and break it into stages.

Stage 1: Select the first letter
Since this letter must be a consonant, and since there are 21 consonants, this stage can be completed in (21 ways).

Stage 2: Select the second letter
Since this letter must be a vowel, and since there are 5 vowels (a, e, i, o, u), this stage can be completed in (5 ways).

Stage 3: Select the third letter
Since this letter must be a consonant, and since we already used 1 consonant for the first letter, there are now 20 consonants remaining.
So, this stage can be completed in (20 ways).

By the Fundamental Counting Principle (FCP) we can complete all 3 stages (and thus create a 3-letter code) in (21)(5)(20) ways ([spoiler]= 2100 ways[/spoiler])

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

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com