permutations

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crackinggmat: Senior | Next Rank: 100 Posts; Posts: 79; Joined: Thu Jun 03, 2010 11:58 am; Thanked: 1 times

by crackinggmat » Tue Oct 11, 2011 10:08 am

Pls help me find my mistake.....

Will must choose a 3 character computer password consisting of 1 letter from the alphabet and 2 distinct digits ,in any order. From how many different passwords can will choose??

A 390
B 2340
C 4680
D 7020
E 14040

My Solution

will can choose from 26 alphabets in 26 ways
he can choose 1st digit in 10 ways and other digit in 9 ways

therefore total number of ways = 26 * 10 * 9

Now he can arrange this combination of letter and digits in 6 ways

hence final answer should is 6* 26*10*9 = 14040

but the actual ans is 7020
this is question from princeton reveiw math workout

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Source: Beat The GMAT — Problem Solving | Tue Oct 11, 2011 10:08 am

shankar.ashwin: Legendary Member; Posts: 966; Joined: Sat Jan 02, 2010 8:06 amFollowed by: 21 members; Thanked: 230 times

by shankar.ashwin » Tue Oct 11, 2011 10:12 am

26 for the alphabets.
10C2 for the digits - 45.

Now this is arranged in 3! ways,

So 26*45*6=7020.

You did 10*9 for numbers, which includes ordering and again multiplied that by 3!. So those would be duplicates.

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n@resh: Senior | Next Rank: 100 Posts; Posts: 61; Joined: Tue Apr 13, 2010 3:29 am; Thanked: 8 times

by n@resh » Tue Oct 11, 2011 10:17 am

crackinggmat wrote:Pls help me find my mistake.....

Will must choose a 3 character computer password consisting of 1 letter from the alphabet and 2 distinct digits ,in any order. From how many different passwords can will choose??

A 390
B 2340
C 4680
D 7020
E 14040

My Solution

will can choose from 26 alphabets in 26 ways
he can choose 1st digit in 10 ways and other digit in 9 ways

therefore total number of ways = 26 * 10 * 9

Now he can arrange this combination of letter and digits in 6 ways

hence final answer should is 6* 26*10*9 = 14040

but the actual ans is 7020
this is question from princeton reveiw math workout

I think the answer is right.

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leonswati: Senior | Next Rank: 100 Posts; Posts: 94; Joined: Sun Aug 29, 2010 3:40 amFollowed by: 1 member; Thanked: 3 times

by leonswati » Tue Oct 11, 2011 10:55 am

crackinggmat wrote:Pls help me find my mistake.....

Will must choose a 3 character computer password consisting of 1 letter from the alphabet and 2 distinct digits ,in any order. From how many different passwords can will choose??

A 390
B 2340
C 4680
D 7020
E 14040

My Solution

will can choose from 26 alphabets in 26 ways
he can choose 1st digit in 10 ways and other digit in 9 ways

therefore total number of ways = 26 * 10 * 9

Now he can arrange this combination of letter and digits in 6 ways

hence final answer should is 6* 26*10*9 = 14040

but the actual ans is 7020
this is question from princeton reveiw math workout

Shankar has explained it really well.... If u still have a doubt look at it this way...

There are three ways to arrange the alphabets :
1) A D D

2) D A D

3) D D A

As u have already Grouped the digits like if there are 10 ways to select the first digit then are are 9 ways to select the second digit .So reordering them again will create duplicates. But as alphabets have not been ordered so there are three ways to position the Alphabets ...

26*3*10*9 = 7020

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Tue Dec 19, 2017 7:21 am

crackinggmat wrote:Pls help me find my mistake.....

Will must choose a 3 character computer password consisting of 1 letter from the alphabet and 2 distinct digits ,in any order. From how many different passwords can will choose??

A 390
B 2340
C 4680
D 7020
E 14040

Since there are 26 letters and 10 digits, the number of 3-character passwords that can be created is 26 x 10 x 9 = 2,340, if the password is in the form of LDD where L means letter and D means digit. However, the password can be also in the form of DLD and DDL, each of which also can be created in 2,340 ways. Thus, the total number of passwords is 2,340 x 3 = 7,020.

Answer: D

Jeffrey Miller
Head of GMAT Instruction
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

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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

Re: permutations

by Brent@GMATPrepNow » Sun Jan 26, 2020 9:08 am

crackinggmat wrote: ↑
Tue Oct 11, 2011 10:08 am
Will must choose a 3 character computer password consisting of 1 letter from the alphabet and 2 distinct digits ,in any order. From how many different passwords can will choose??

A 390
B 2340
C 4680
D 7020
E 14040

Take the task of creating a password and break it into stages.

Stage 1: Select the one letter to be used in the code
There are 26 letters from which to choose, so we can complete this stage in 26 ways.

Stage 2: Select the two digits to be used in the code
Since the order in which we select the two digits does not matter, we can use combinations.
We can select 2 digits from 10 women in 10C2 ways (45 ways)
So, we can complete stage 2 in 45 ways

NOTE: We now have the 3 characters to be used in the code. At this point, we need to arrange those 3 characters.

Stage 3: Arrange the 3 selected characters.
RULE: We can arrange n unique objects in n! ways.
So, we can arrange the 3 characters in 3! ways (6 ways)
So we can complete stage 3 in 6 ways.

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create a password ) in (26)(45)(6) ways
(26)(45)(6) = 7020

Answer: D

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

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat ... /video/776

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

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/permutation- ... 73915.html
- https://www.beatthegmat.com/permutation-t122873.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/combinations-t123249.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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