Last visit was: 22 Jun 2025, 03:57 It is currently 22 Jun 2025, 03:57
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
User avatar
RadhaKrishnan
Joined: 19 Nov 2011
Last visit: 10 May 2013
Posts: 4
Own Kudos:
1,119
 [158]
Posts: 4
Kudos: 1,119
 [158]
5
Kudos
Add Kudos
153
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Jun 2025
Posts: 102,227
Own Kudos:
Given Kudos: 93,968
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,227
Kudos: 734,439
 [45]
13
Kudos
Add Kudos
32
Bookmarks
Bookmark this Post
User avatar
eaakbari
Joined: 24 Mar 2010
Last visit: 18 Nov 2013
Posts: 46
Own Kudos:
418
 [17]
Given Kudos: 133
Posts: 46
Kudos: 418
 [17]
14
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
General Discussion
User avatar
mbaiseasy
Joined: 13 Aug 2012
Last visit: 29 Dec 2013
Posts: 323
Own Kudos:
1,977
 [5]
Given Kudos: 11
Concentration: Marketing, Finance
GPA: 3.23
Posts: 323
Kudos: 1,977
 [5]
4
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
RadhaKrishnan
If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

A. 5 to 4
B. 3 to 2
C. 2 to 1
D. 5 to 1
E. 6 to 1

Number of 5-letter code formed from 10 letters: \(=10*9*8*7*6\)
Number of 4-letter code formed from 10 letters: \(=10*9*8*7\)

Answer: 6 to 1 or E
User avatar
mbaiseasy
Joined: 13 Aug 2012
Last visit: 29 Dec 2013
Posts: 323
Own Kudos:
1,977
 [2]
Given Kudos: 11
Concentration: Marketing, Finance
GPA: 3.23
Posts: 323
Kudos: 1,977
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Orange08
If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

A. 5 to 4
B. 3 to 2
C. 2 to 1
D. 5 to 1
E. 6 to 1

Number of ways to form 5-letter code: 10!/5! = 10*9*8*7*6
Number of ways to form 4-letter code: 10!/6! = 10*9*8*7

Ratio: 6 to 1

Answer : E
User avatar
mydreammba
Joined: 28 Jul 2011
Last visit: 06 Dec 2013
Posts: 224
Own Kudos:
Given Kudos: 16
Location: United States
Concentration: International Business, General Management
GPA: 3.86
WE:Accounting (Commercial Banking)
Posts: 224
Kudos: 1,569
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
RadhaKrishnan
If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

A. 5 to 4
B. 3 to 2
C. 2 to 1
D. 5 to 1
E. 6 to 1

Notice that as we are dealing with code words then the order of the letters matters.

# of ways to choose 5 different letters out of given 10 letters when the order of chosen letters matters (as for example code word ABCDE is different from BCDEA) is \(P^5_{10}\);

# of ways to choose 4 different letters out of given 10 letters when the order of chosen letters matters is \(P^4_{10}\);

\(Ratio=\frac{P^5_{10}}{P^4_{10}}=\frac{10!}{5!}*\frac{6!}{10!}=\frac{6}{1}\).

Answer: E.

Hi Bunnel,

In this problem you have used Permutations, but in the problem you have used combination, which also deals with code

a-researcher-plans-to-identify-each-participant-in-a-certain-134584.html

Can you please when to use permutaions or Combinations in these type of problems?
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Jun 2025
Posts: 102,227
Own Kudos:
734,439
 [4]
Given Kudos: 93,968
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,227
Kudos: 734,439
 [4]
1
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
mydreammba
Bunuel
RadhaKrishnan
If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

A. 5 to 4
B. 3 to 2
C. 2 to 1
D. 5 to 1
E. 6 to 1

Notice that as we are dealing with code words then the order of the letters matters.

# of ways to choose 5 different letters out of given 10 letters when the order of chosen letters matters (as for example code word ABCDE is different from BCDEA) is \(P^5_{10}\);

# of ways to choose 4 different letters out of given 10 letters when the order of chosen letters matters is \(P^4_{10}\);

\(Ratio=\frac{P^5_{10}}{P^4_{10}}=\frac{10!}{5!}*\frac{6!}{10!}=\frac{6}{1}\).

Answer: E.

Hi Bunnel,

In this problem you have used Permutations, but in the problem you have used combination, which also deals with code

a-researcher-plans-to-identify-each-participant-in-a-certain-134584.html

Can you please when to use permutaions or Combinations in these type of problems?

In this case the order of the letters matters, but in other question we are only interested in codes which are in alphabetical order (so we are interested in only one particular order).

This post might help: a-researcher-plans-to-identify-each-participant-in-a-certain-134584.html#p1150091
User avatar
Tmoni26
User avatar
LBS Moderator
Joined: 13 Jan 2015
Last visit: 10 Aug 2017
Posts: 88
Own Kudos:
Given Kudos: 67
Location: United Kingdom
Concentration: Other, General Management
GMAT 1: 690 Q48 V36
GMAT 1: 690 Q48 V36
Posts: 88
Kudos: 49
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello Bunuel,

Thank you very much for responding.

My problem with your initial calculation is that when i do understand how you got 6! in the bottom part of your example above.

When i calculate it, it goes 10 Choose 5 is = 10*9*8*7*6 / 5*4*3*2*1

And then 10 Choose 4 i= 10*9*8*7 / 4*3*2*1

when I calculate 10 Choose 5 / 10 Choose 4.... I get 6/5

I do not know where the error is coming from

Thanks alot
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Jun 2025
Posts: 102,227
Own Kudos:
Given Kudos: 93,968
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,227
Kudos: 734,439
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Tmoni26
Hello Bunuel,

Thank you very much for responding.

My problem with your initial calculation is that when i do understand how you got 6! in the bottom part of your example above.

When i calculate it, it goes 10 Choose 5 is = 10*9*8*7*6 / 5*4*3*2*1

And then 10 Choose 4 i= 10*9*8*7 / 4*3*2*1

when I calculate 10 Choose 5 / 10 Choose 4.... I get 6/5

I do not know where the error is coming from

Thanks alot

Please use correct notations...

\(P^5_{10}=\frac{10!}{5!}\)

\(P^4_{10}=\frac{10!}{6!}\)

\(\frac{P^5_{10}}{P^4_{10}}=\frac{10!}{5!}*\frac{6!}{10!}=\frac{6}{1}\).

Theory on Combinations: math-combinatorics-87345.html
User avatar
SVaidyaraman
Joined: 17 Dec 2012
Last visit: 09 Jun 2025
Posts: 577
Own Kudos:
1,729
 [1]
Given Kudos: 20
Location: India
Expert
Expert reply
Posts: 577
Kudos: 1,729
 [1]
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Take 1 case: ABCDE for 5 letter code and ABCD for 4 letter code. You Choose 5 at a time and 4 at a time and order is important. So the formula is nPr. Therefore the ratio is 10P5/ 10P4 = 6:1
Hence E.
User avatar
EMPOWERgmatRichC
User avatar
Major Poster
Joined: 19 Dec 2014
Last visit: 31 Dec 2023
Posts: 21,789
Own Kudos:
12,445
 [1]
Given Kudos: 450
Status:GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Expert
Expert reply
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Posts: 21,789
Kudos: 12,445
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi All,

To calculate the number of 5-letter codes and 4-letter codes, we have to set up 2 permutations. There's a 'shortcut' though - since the answer choices are RATIOS, we don't actually have to calculate the total number of each type of code.

Total 5-letter codes = (10)(9)(8)(7)(6)
Total 4-letter codes = (10)(9)(8)(7)

Notice how the number of 5-letter codes is the total of 4-letter codes multiplied by 6. Thus, the ratio of codes is 6:1

Final Answer:
GMAT assassins aren't born, they're made,
Rich
avatar
panache67
Joined: 25 Feb 2018
Last visit: 12 Jul 2018
Posts: 1
Given Kudos: 92
Posts: 1
Kudos: 0
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
RadhaKrishnan
If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

A. 5 to 4
B. 3 to 2
C. 2 to 1
D. 5 to 1
E. 6 to 1

Notice that as we are dealing with code words then the order of the letters matters.

# of ways to choose 5 different letters out of given 10 letters when the order of chosen letters matters (as for example code word ABCDE is different from BCDEA) is \(P^5_{10}\);

# of ways to choose 4 different letters out of given 10 letters when the order of chosen letters matters is \(P^4_{10}\);

Answer: E.


Hi Bunuel,

I struggle with the the order of \(P^5_{10}\). From the theory I understood that the permutation is defined as: \(P^n_{k}\) n being the set of distinct objects and k being the number of objects chosen. Why is this upside down?

Thanks!
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Jun 2025
Posts: 102,227
Own Kudos:
734,439
 [1]
Given Kudos: 93,968
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,227
Kudos: 734,439
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
panache67
Bunuel
RadhaKrishnan
If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

A. 5 to 4
B. 3 to 2
C. 2 to 1
D. 5 to 1
E. 6 to 1

Notice that as we are dealing with code words then the order of the letters matters.

# of ways to choose 5 different letters out of given 10 letters when the order of chosen letters matters (as for example code word ABCDE is different from BCDEA) is \(P^5_{10}\);

# of ways to choose 4 different letters out of given 10 letters when the order of chosen letters matters is \(P^4_{10}\);

Answer: E.


Hi Bunuel,

I struggle with the the order of \(P^5_{10}\). From the theory I understood that the permutation is defined as: \(P^n_{k}\) n being the set of distinct objects and k being the number of objects chosen. Why is this upside down?

Thanks!

\(P^5_{10}\), \(P^{10}_5\), 10P5 are all the same: choosing 5 out of 10, when the order matters. Just different ways of writing the same. Could it be choosing 10 out of 5?
User avatar
CEdward
Joined: 11 Aug 2020
Last visit: 14 Apr 2022
Posts: 1,212
Own Kudos:
Given Kudos: 332
Posts: 1,212
Kudos: 244
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Permutation problem without repeats.

5-letter word permutations: 10 x 9 x 8 x 7 x 6
4-letter word permutations: 10 x 9 x 8 x 7

5-letter:4-letter = 6:1

Answer is E.
User avatar
BrentGMATPrepNow
User avatar
Major Poster
Joined: 12 Sep 2015
Last visit: 13 May 2024
Posts: 6,757
Own Kudos:
33,889
 [2]
Given Kudos: 799
Location: Canada
Expert
Expert reply
Posts: 6,757
Kudos: 33,889
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
RadhaKrishnan
If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

A. 5 to 4
B. 3 to 2
C. 2 to 1
D. 5 to 1
E. 6 to 1

Although we could use the permutation formula to answer this question, we can also solve using the Fundamental Counting Principle (FCP, aka the slot method). In fact, we can solve any permutation question using the FCP.

Number of 5-letter words we can make
We can select the 1st letter in 10 ways
We can select the 2nd letter in 9 ways
We can select the 3rd letter in 8 ways
We can select the 4th letter in 7 ways
We can select the 5th letter in 6 ways
So the total number of ways to construct a 5-letter word = (10)(9)(8)(7)(6)

Number of 4-letter words we can make
We can select the 1st letter in 10 ways
We can select the 2nd letter in 9 ways
We can select the 3rd letter in 8 ways
We can select the 4th letter in 7 ways
So the total number of ways to construct a 4-letter word = (10)(9)(8)(7)

Ratio of the number of 5-letter code words to the number of 4-letter code words = (10)(9)(8)(7)(6)/(10)(9)(8)(7) = 6/1 = 6 to 1

Answer: E

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

RELATED VIDEO
User avatar
gmatchile
Joined: 26 Jan 2010
Last visit: 14 Nov 2023
Posts: 43
Own Kudos:
Given Kudos: 1
Status:casado
Location: chile
Concentration: Educación
WE 1: <!-- m --><a class="postlink" href=""></a><!-- m -->
WE 2: <!-- m --><a class="postlink" href=""></a><!-- m -->
WE 3: <!-- m --><a class="postlink" href=""></a><!-- m -->
Posts: 43
Kudos: 18
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Of 10 different possibilities.

(I choose 5, different)/(I choose 4, different)

(E1 E2 E3 E4 E5)/(E1 E2 E3 E4)

In the numerator we have 5 spaces and in the denominator 4 spaces.

Each space must be filled with the 10 different letters indicated, keeping in mind that the letters must be different.

In both the numerator and the denominator, E1 can be filled by any of the 10 letters, E2 can be filled by only 9 of the letters (it already occupied a letter in E1, since all the letters must be different) and so on.

(10x9x8x7x6)/(10x9x8x7)
6/1

Answer E
User avatar
GmatPoint
Joined: 02 Jan 2022
Last visit: 13 Oct 2022
Posts: 248
Own Kudos:
Given Kudos: 3
GMAT 1: 760 Q50 V42
GMAT 1: 760 Q50 V42
Posts: 248
Kudos: 131
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The total number of 5 codes from the 10 letters = 10C5*5!

The total number of 4 letter codes from 10 letters = 10C4*4!

The required ratio = 10C5*5!/10C4*4! = 6

Thus, the correct option is E.
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 37,248
Own Kudos:
Posts: 37,248
Kudos: 1,002
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Moderators:
Math Expert
102227 posts
PS Forum Moderator
654 posts