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sana.noor: Legendary Member; Posts: 512; Joined: Mon Jun 18, 2012 11:31 pmFollowed by: 20 members; Thanked: 42 times

by sana.noor » Sun Jul 21, 2013 12:39 pm

How many 3 letter codes can be formed from the PEPPER with replacement.

a)6
b)18
c)19
d)20
e)36

OA is C

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Source: Beat The GMAT — Problem Solving | Sun Jul 21, 2013 12:39 pm

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

by Brent@GMATPrepNow » Sun Jul 21, 2013 12:59 pm

sana.noor wrote:How many 3 letter codes can be formed from the PEPPER with replacement.

a)6
b)18
c)19
d)20
e)36

OA is C

There other other counting techniques we could use here, but the relatively small answer choices suggest that listing/counting might be the fastest approach.

Let's examine the different cases.

All 3 letters the same
Only possible with three P's
PPP (1 code)

2 letters the same, 1 different
- 2 P's and 1 R
RPP, PRP, PPR
(3 codes)

- 2 P's and 1 E
(3 codes)

- 2 E's and 1 P
(3 codes)

- 2 E's and 1 R
(3 codes)

All 3 letters different (1 P, 1 R and 1 E)
We can arrange 3 unique objects in 3! ways (6 ways)
(6 codes)

Add them up: 1+3+3+3+3+6 = [spoiler]19 = C[/spoiler]

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pmFollowed by: 4 members; Thanked: 29 times

by vipulgoyal » Mon Jul 22, 2013 3:43 am

3p = 1, 2P1R = 3,2P1E= 3, 2E1R = 3, 2E1P =3,1P1R1E =6

3+3+3+3+1+6 = 19

Last edited by vipulgoyal on Tue Jul 23, 2013 12:39 am, edited 1 time in total.

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satish_iitg: Senior | Next Rank: 100 Posts; Posts: 43; Joined: Sat May 19, 2007 10:38 amFollowed by: 1 member; Thanked: 7 times; GMAT Score:690

by satish_iitg » Mon Jul 22, 2013 9:19 am

Brent@GMATPrepNow wrote:
sana.noor wrote:How many 3 letter codes can be formed from the PEPPER with replacement.

a)6
b)18
c)19
d)20
e)36

OA is C
There other other counting techniques we could use here, but the relatively small answer choices suggest that listing/counting might be the fastest approach.

Let's examine the different cases.

All 3 letters the same
Only possible with three P's
PPP (1 code)

2 letters the same, 1 different
- 2 P's and 1 R
RPP, PRP, PPR
(3 codes)

- 2 P's and 1 E
(3 codes)

- 2 E's and 1 P
(3 codes)

- 2 E's and 1 R
(3 codes)

All 3 letters different (1 P, 1 R and 1 E)
We can arrange 3 unique objects in 3! ways (6 ways)
(6 codes)

Add them up: 1+3+3+3+3+6 = [spoiler]19 = C[/spoiler]

Cheers,
Brent

Dear Brent,

Can you please tell me why 6C3 is wrong.

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

by Brent@GMATPrepNow » Mon Jul 22, 2013 9:40 am

satish_iitg wrote:
Dear Brent,

Can you please tell me why 6C3 is wrong.

There are two reasons why 6C3 (combinations) won't work here.

First, since the order of the 3 letters matters (e.g., the word PRE is different from the word REP) we cannot use combinations. We use combinations when the order does not matter (for example, we can select 3 friends from 6 friends in 6C3 ways).

Second, we have duplicated letters (three P's and two E's). To illustrate why duplicated letters messes things up, let's examine what would happen if the question were, "How many 3 letter codes can be formed from PPPPPP?" Here, we can see that 6C3 won't work because all of the letters are the same. In fact, there's only one 3-letter word we can create (PPP). In the same way, the duplicated letters in PEPPER make things trickier.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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sarawutc: Newbie | Next Rank: 10 Posts; Posts: 5; Joined: Sun Mar 31, 2013 10:26 am

by sarawutc » Mon Jul 22, 2013 9:46 am

I have a question... the question says "with replacement". Doesn't it mean that you can have 3Ps, 3Es, and 3Rs. I'm so confused.

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

by Brent@GMATPrepNow » Mon Jul 22, 2013 9:59 am

sarawutc wrote:I have a question... the question says "with replacement". Doesn't it mean that you can have 3Ps, 3Es, and 3Rs. I'm so confused.

Yeah, I didn't like that phrase, so I just ignored it

I have no idea what the author of the question intended, but you can be certain that an official GMAT question would never contain such ambiguity.

The question, as I read it, is asking asking us to choose 3 letters from PEPPER. So, for example, we can't have three R's.

If the question intended us to be able to use up to 3Ps, 3Es, and 3Rs, then it doesn't make any sense to mention "PEPPER." The question should have just asked, "If duplication is permitted, how many 3-letter codes can be formed using only P's, R's and E's?"

Of course, that still may have been the author's intention. We can't be sure.

Let's answer the question, "If duplication is permitted, how many 3-letter codes can be formed using only P's, R's and E's?"

Take the task of building 3-letter codes and break it into stages.

Stage 1: Select the first letter
We have 3 letters to choose from (P, E, or R), so we can accomplish stage 1 in 3 ways.

Stage 2: Select the second letter
We have 3 letters to choose from (P, E, or R), so we can accomplish stage 2 in 3 ways.

Stage 3: Select the third letter
We have 3 letters to choose from (P, E, or R), so we can accomplish stage 3 in 3 ways.

By the Fundamental Counting Principle (FCP) we can complete all 3 stages (and thus create a 3-letter code) in (3)(3)(3) ways ([spoiler]= 27 ways[/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

Brent Hanneson - Creator of GMATPrepNow.com

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satish_iitg: Senior | Next Rank: 100 Posts; Posts: 43; Joined: Sat May 19, 2007 10:38 amFollowed by: 1 member; Thanked: 7 times; GMAT Score:690

by satish_iitg » Mon Jul 22, 2013 10:48 am

Thanks a lot Brent.

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gocoder: Master | Next Rank: 500 Posts; Posts: 120; Joined: Sat Dec 05, 2015 11:09 am

by gocoder » Sun Dec 04, 2016 4:07 am

Brent@GMATPrepNow wrote:How many 3 letter codes can be formed from the PEPPER with replacement.

a)6
b)18
c)19
d)20
e)36

OA is C

We can arrange 3 unique objects in 3! ways (6 ways)

Cheers,
Brent[/quote]

If the Same question had asked to pick 3 letters without replacing, whether the answer would be 6 ?

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

by Brent@GMATPrepNow » Sun Dec 04, 2016 2:42 pm

gocoder wrote:
Brent@GMATPrepNow wrote:How many 3 letter codes can be formed from the PEPPER with replacement.

a)6
b)18
c)19
d)20
e)36

OA is C
We can arrange 3 unique objects in 3! ways (6 ways)

Cheers,
Brent

If the Same question had asked to pick 3 letters without replacing, whether the answer would be 6 ?[/quote]

That alteration to the question would really complicate matters, enough to make it out of scope for the GMAT.
The answer would be more than 6 though (using just P's and E's, there are 7 possible arrangements).

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Thu Dec 08, 2016 8:36 pm

Brent@GMATPrepNow wrote:Yeah, I didn't like that phrase, so I just ignored it
I have no idea what the author of the question intended, but you can be certain that an official GMAT question would never contain such ambiguity.

Seconded! It's strange.

If I had to guess, I'd say it means something like this. Imagine that the six letters P, E, P, P, E, and R are in a jar. You're going to pick one out, write it down, then put that letter back in the jar. You're then going to pick again, write that letter down, etc.

So something like EEP or ERR or EEE would be a valid code (... and a valid response to the prompt

)

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