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How many 4 digit codes can be made, if each code can only contain
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Updated on: 09 May 2017, 08:48
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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.
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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.



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Re: How many 4 digit codes can be made, if each code can only contain
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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.
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Re: How many 4 digit codes can be made, if each code can only contain
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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
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Re: How many 4 digit codes can be made, if each code can only contain
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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???



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Re: How many 4 digit codes can be made, if each code can only contain
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05 Oct 2010, 13:37



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Re: How many 4 digit codes can be made, if each code can only contain
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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...



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



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Re: How many 4 digit codes can be made, if each code can only contain
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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
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Re: How many 4 digit codes can be made, if each code can only contain
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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




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