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

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How many 4 digit codes can be made, if each code can only [#permalink]

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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.
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Jul 2013, 07:44, edited 1 time in total.
RENAMED THE TOPIC.
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Re: 4 digit codes [#permalink]

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New post 04 Oct 2010, 07:41
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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 4-digit number could be of the following type:

OOOO, for example: 2|3|5|7 or 2|2|7|7. Each O can take 4 values from {2, 3, 5, 7}, so total ways for this type is \(4^4\);

TT, for example: 11|11 or 19|17. Each T can take 4 values from {11, 13, 17, 19}, so total ways for this type is \(4^2\);

TOO, for example: 11|3|5 or 19|7|2. 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: 2|13|5 or 7|19|2. The same as above: \(4^3\);
OOT, for example: 2|5|19 or 7|2|17. The same as above: \(4^3\);

Total: \(4^4+4^2+3*4^3=464\).

Answer: C.
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Re: 4 digit codes [#permalink]

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New post 04 Oct 2010, 07:45
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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 2-digit ones {11,13,17,19}

Case 1
Codes formed with 2 two digit primes
(2-digit prime) (2-digit prime)
No of ways = 4x4 = 16

Case 2
Codes formed with 4 one digit primes
(1-digit prime) (1-digit prime) (1-digit prime) (1-digit prime)
No of ways = 4x4x4x4 = 256

Case 3
Codes formed with 2 one-digit primes and 1 two-digit prime
(1-digit prime) (1-digit prime) (2-digit prime)
(1-digit prime) (2-digit prime) (1-digit prime)
(2-digit prime) (1-digit prime) (1-digit 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 :wink:
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Re: 4 digit codes [#permalink]

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New post 04 Oct 2010, 08:41
Hi guys,
Thanks you BOTH for the explanation. It is clear now.
You got +1 from me.
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Re: 4 digit codes [#permalink]

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New post 05 Oct 2010, 12:33
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 4-digit number could be of the following type:

OOOO, for example: 2|3|5|7 or 2|2|7|7. Each O can take 4 values from {2, 3, 5, 7}, so total ways for this type is \(4^4\);

TT, for example: 11|11 or 19|17. Each T can take 4 values from {11, 13, 17, 19}, so total ways for this type is \(4^2\);

TOO, for example: 11|3|5 or 19|7|2. 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: 2|13|5 or 7|19|2. The same as above: \(4^3\);
OOT, for example: 2|5|19 or 7|2|17. 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: 4 digit codes [#permalink]

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New post 05 Oct 2010, 12:37
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.
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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: 4 digit codes [#permalink]

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New post 05 Oct 2010, 12:53
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: 4 digit codes [#permalink]

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New post 05 Oct 2010, 13:01
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Expert's post
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 1-digit primes can be the same, but it's not important here.

We are counting # of ways 4-digit number can be formed with two 1-digit primes and one 2-digit prime:
{1-digit}{1-digit}{2-digit}
{1-digit}{2-digit}{1-digit}
{2-digit}{1-digit}{1-digit}

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: 4 digit codes [#permalink]

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New post 05 Oct 2010, 13:59
Thanks Bunuel... +1 ... u already have many I knw :)
Re: 4 digit codes   [#permalink] 05 Oct 2010, 13:59
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