Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 01 Sep 2010
Posts: 20

How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
Updated on: 09 May 2017, 08:48
Question Stats:
42% (02:47) correct 58% (01:50) wrong based on 79 sessions
HideShow timer Statistics
How many 4 digit codes can be made, if each code can only contain prime numbers that are less than 20? A. 24 B. 102 C. 464 D. 656 E. 5040 Hi all, As you saw, I have published a bunch of questions in the past hour. Most of these questions are taken from a collection of hard quantitative questions provided by The Princeton Review (a.k.a  Killer Math). I have posted any question that I have solved incorrectly, either due to careless error or concept error, in order to share them with everyone here. Thank you all for the help and detailed explanations (especially you  Bunuel). It is very helpful.
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by eladshush on 04 Oct 2010, 07:00.
Last edited by Bunuel on 09 May 2017, 08:48, edited 2 times in total.
RENAMED THE TOPIC.



Math Expert
Joined: 02 Sep 2009
Posts: 47200

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
04 Oct 2010, 08:41
eladshush wrote: Hi all,
As you saw, I have published a bunch of questions in the past hour. Most of these questions are taken from a collection of hard quantitative questions provided by The Princeton Review (a.k.a  Killer Math).
I have posted any question that I have solved incorrectly, either due to careless error or concept error, in order to share them with everyone here.
Please consider the following problem that I am not sure I understood:
How many 4 digit codes can be made, if each code can only contain prime numbers that are less than 20?
A. 24 B. 102 C. 464 D. 656 E. 5040
Thank you all for the help and detailed explanations (especially you  Bunuel). It is very helpful. The question is a little bit ambiguous but I think it means the following: I guess as it's not mentioned primes can be repeated. There are: 4 one digit primes (O) less than 20  2, 3, 5, 7; 4 two digit primes (T) less than 20  11, 13, 17, 19; Thus 4digit number could be of the following type: OOOO, for example: 2357 or 2277. Each O can take 4 values from {2, 3, 5, 7}, so total ways for this type is \(4^4\); TT, for example: 1111 or 1917. Each T can take 4 values from {11, 13, 17, 19}, so total ways for this type is \(4^2\); TOO, for example: 1135 or 1972. T can take 4 values from {11, 13, 17, 19} and each O can take 4 values from {2, 3, 5, 7}, so total ways for this type is \(4*4^2=4^3\); OTO, for example: 2135 or 7192. The same as above: \(4^3\); OOT, for example: 2519 or 7217. The same as above: \(4^3\); Total: \(4^4+4^2+3*4^3=464\). Answer: C.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Retired Moderator
Joined: 02 Sep 2010
Posts: 775
Location: London

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
04 Oct 2010, 08:45
eladshush wrote: Hi all,
As you saw, I have published a bunch of questions in the past hour. Most of these questions are taken from a collection of hard quantitative questions provided by The Princeton Review (a.k.a  Killer Math).
I have posted any question that I have solved incorrectly, either due to careless error or concept error, in order to share them with everyone here.
Please consider the following problem that I am not sure I understood:
How many 4 digit codes can be made, if each code can only contain prime numbers that are less than 20?
A. 24 B. 102 C. 464 D. 656 E. 5040
Thank you all for the help and detailed explanations (especially you  Bunuel). It is very helpful. First note all the single digit primes {2,3,5,7} And then all the 2digit ones {11,13,17,19} Case 1Codes formed with 2 two digit primes (2digit prime) (2digit prime) No of ways = 4x4 = 16 Case 2Codes formed with 4 one digit primes (1digit prime) (1digit prime) (1digit prime) (1digit prime) No of ways = 4x4x4x4 = 256 Case 3Codes formed with 2 onedigit primes and 1 twodigit prime (1digit prime) (1digit prime) (2digit prime) (1digit prime) (2digit prime) (1digit prime) (2digit prime) (1digit prime) (1digit prime) Each set can be formed in 4x4x4 ways So total = 3x64 = 192 Total number = 192+256+16 = 464 Answer is (c)PS : Some others also deserve thanks
_________________
Math writeups 1) Algebra101 2) Sequences 3) Set combinatorics 4) 3D geometry
My GMAT story
GMAT Club Premium Membership  big benefits and savings



Manager
Joined: 27 Mar 2010
Posts: 100

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
05 Oct 2010, 13:33
Bunuel wrote: eladshush wrote: Hi all,
As you saw, I have published a bunch of questions in the past hour. Most of these questions are taken from a collection of hard quantitative questions provided by The Princeton Review (a.k.a  Killer Math).
I have posted any question that I have solved incorrectly, either due to careless error or concept error, in order to share them with everyone here.
Please consider the following problem that I am not sure I understood:
How many 4 digit codes can be made, if each code can only contain prime numbers that are less than 20?
A. 24 B. 102 C. 464 D. 656 E. 5040
Thank you all for the help and detailed explanations (especially you  Bunuel). It is very helpful. The question is a little bit ambiguous but I think it means the following: I guess as it's not mentioned primes can be repeated. There are: 4 one digit primes (O) less than 20  2, 3, 5, 7; 4 two digit primes (T) less than 20  11, 13, 17, 19; Thus 4digit number could be of the following type: OOOO, for example: 2357 or 2277. Each O can take 4 values from {2, 3, 5, 7}, so total ways for this type is \(4^4\); TT, for example: 1111 or 1917. Each T can take 4 values from {11, 13, 17, 19}, so total ways for this type is \(4^2\); TOO, for example: 1135 or 1972. T can take 4 values from {11, 13, 17, 19} and each O can take 4 values from {2, 3, 5, 7}, so total ways for this type is \(4*4^2=4^3\); OTO, for example: 2135 or 7192. The same as above: \(4^3\); OOT, for example: 2519 or 7217. The same as above: \(4^3\); Total: \(4^4+4^2+3*4^3=464\). Answer: C. Hi Bunuel, why can't i write TOO,OTO,OOT AS (4^3)*3! , taking the T as one entity ans assuming that 3 things can be arranged in 3! ways???



Math Expert
Joined: 02 Sep 2009
Posts: 47200

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
05 Oct 2010, 13:37



Manager
Joined: 27 Mar 2010
Posts: 100

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
05 Oct 2010, 13:53
Bunuel wrote: utin wrote: Hi Bunuel,
why can't i write TOO,OTO,OOT AS
(4^3)*3! , taking the T as one entity ans assuming that 3 things can be arranged in 3! ways??? Just one thing: TOO can be arranged in 3!/2! ways and not in 3! (# of permutations of 3 letters out which 2 O's are identical is 3!/2!), so it would be \(4^3*\frac{3!}{2!}=4^3*3\). Hope it's clear. I though about the same but but when i see that TOO as three things to be arranged in 3! ways then i also thought that OO ARE TWO DIGITS AND THEY ARE TWO DIFFERENT PRIME NOS SO WHY DIVIDE BY 2! this might clear my entire probability confusion i hope...



Math Expert
Joined: 02 Sep 2009
Posts: 47200

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
05 Oct 2010, 14:01
utin wrote: Bunuel wrote: utin wrote: Hi Bunuel,
why can't i write TOO,OTO,OOT AS
(4^3)*3! , taking the T as one entity ans assuming that 3 things can be arranged in 3! ways??? Just one thing: TOO can be arranged in 3!/2! ways and not in 3! (# of permutations of 3 letters out which 2 O's are identical is 3!/2!), so it would be \(4^3*\frac{3!}{2!}=4^3*3\). Hope it's clear. I though about the same but but when i see that TOO as three things to be arranged in 3! ways then i also thought that OO ARE TWO DIGITS AND THEY ARE TWO DIFFERENT PRIME NOS SO WHY DIVIDE BY 2! this might clear my entire probability confusion i hope... First of all two 1digit primes can be the same, but it's not important here. We are counting # of ways 4digit number can be formed with two 1digit primes and one 2digit prime: {1digit}{1digit}{2digit} {1digit}{2digit}{1digit} {2digit}{1digit}{1digit} Total 3 ways.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 27 Mar 2010
Posts: 100

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
05 Oct 2010, 14:59
Thanks Bunuel... +1 ... u already have many I knw



Director
Joined: 13 Mar 2017
Posts: 610
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
03 Oct 2017, 00:09
eladshush wrote: How many 4 digit codes can be made, if each code can only contain prime numbers that are less than 20?
A. 24 B. 102 C. 464 D. 656 E. 5040
Hi all,
As you saw, I have published a bunch of questions in the past hour. Most of these questions are taken from a collection of hard quantitative questions provided by The Princeton Review (a.k.a  Killer Math).
I have posted any question that I have solved incorrectly, either due to careless error or concept error, in order to share them with everyone here.
Thank you all for the help and detailed explanations (especially you  Bunuel).
It is very helpful. Though the question is little bit confusing, but conceptually this question is a real gem. More details should have been provided about the restrictions on selection of digits. So, there are 4 one digit prime numbers to be used they are {2,3,5,7}. Lets call the single digits as O & there are 4 2digit prime numbers to be used they are {11, 13, 15, 17}. And double digits as T. Now there are 3 ways of selecting digits. Case 1: All one digit prime numbers OOOO No. of codes = 4^4 (since repetition of digit is allowed = 256 Case 2: 2 Two digit prime numbers TT No. of codes = 4*4 = 16 Case 3 : one two digit prime number & 2 one digit prime number : OOT, OTO, TOO No. of codes = 4*4*4 + 4*4*4 + 4*4*4 = 3*4^3 = 192 Total no. of codes = 256 + 16 + 192 = 464 Answer C
_________________
CAT 99th percentiler : VA 97.27  DILR 96.84  QA 98.04  OA 98.95 UPSC Aspirants : Get my app UPSC Important News Reader from Play store.
MBA Social Network : WebMaggu
Appreciate by Clicking +1 Kudos ( Lets be more generous friends.) What I believe is : "Nothing is Impossible, Even Impossible says I'm Possible" : "Stay Hungry, Stay Foolish".



Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 278

Re: How many 4 digit codes can be made, if each code can only contain
[#permalink]
Show Tags
25 Dec 2017, 06:59
Bunuel wrote: eladshush wrote: Hi all,
As you saw, I have published a bunch of questions in the past hour. Most of these questions are taken from a collection of hard quantitative questions provided by The Princeton Review (a.k.a  Killer Math).
I have posted any question that I have solved incorrectly, either due to careless error or concept error, in order to share them with everyone here.
Please consider the following problem that I am not sure I understood:
How many 4 digit codes can be made, if each code can only contain prime numbers that are less than 20?
A. 24 B. 102 C. 464 D. 656 E. 5040
Thank you all for the help and detailed explanations (especially you  Bunuel). It is very helpful. The question is a little bit ambiguous but I think it means the following: I guess as it's not mentioned primes can be repeated. There are: 4 one digit primes (O) less than 20  2, 3, 5, 7; 4 two digit primes (T) less than 20  11, 13, 17, 19; Thus 4digit number could be of the following type: OOOO, for example: 2357 or 2277. Each O can take 4 values from {2, 3, 5, 7}, so total ways for this type is \(4^4\); TT, for example: 1111 or 1917. Each T can take 4 values from {11, 13, 17, 19}, so total ways for this type is \(4^2\); TOO, for example: 1135 or 1972. T can take 4 values from {11, 13, 17, 19} and each O can take 4 values from {2, 3, 5, 7}, so total ways for this type is \(4*4^2=4^3\); OTO, for example: 2135 or 7192. The same as above: \(4^3\); OOT, for example: 2519 or 7217. The same as above: \(4^3\); Total: \(4^4+4^2+3*4^3=464\). Answer: C. great solution as always, man




Re: How many 4 digit codes can be made, if each code can only contain &nbs
[#permalink]
25 Dec 2017, 06:59






