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# If a code word is defined to be a sequence of different

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If a code word is defined to be a sequence of different [#permalink]

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Updated on: 28 Jan 2012, 04:02
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69% (01:05) correct 31% (01:21) wrong based on 871 sessions

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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

Originally posted by RadhaKrishnan on 28 Jan 2012, 03:59.
Last edited by Bunuel on 28 Jan 2012, 04:02, edited 1 time in total.
Added the OA
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Re: Ratio [#permalink]

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28 Jan 2012, 04:02
6
7
RadhaKrishnan wrote:
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.
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Re: If a code word is defined to be a sequence of different [#permalink]

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23 Dec 2012, 13:29
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1
Numbers of Options applicable for 5 letter digit -> $$10 * 9 * 8 * 7 * 6$$
( as option pool for first digit is 10, for second 9 because one is removed and so on)
Numbers of Options applicable for 5 letter digit -> $$10 * 9 * 8 * 7$$

Required Ratio -> $$(10 * 9 *8 * 7 * 6)/(10 * 9* 8 * 7)$$ = $$6:1$$
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Re: If a code word is defined to be a sequence of different [#permalink]

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28 Dec 2012, 06:43
2
1
RadhaKrishnan wrote:
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
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Re: Ratio [#permalink]

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30 Mar 2013, 09:04
Bunuel wrote:
RadhaKrishnan wrote:
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?
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Re: Ratio [#permalink]

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31 Mar 2013, 08:51
mydreammba wrote:
Bunuel wrote:
RadhaKrishnan wrote:
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
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Re: If a code word is defined to be a sequence of different [#permalink]

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23 Aug 2014, 10:51
Bunuel wrote:
RadhaKrishnan wrote:
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 Bunuel,

A couple of clarifications as i'm going through similar problems:

1) In this case, we use permutations because order DOES NOT matter. Correct?
2) If the problem said that we had to use the letter in alphabetical order, then order would matter and we would have to use the Combination formula over permutation. Correct?
3) Permutation and Combination both assumes that the letters/numbers CANNOT be repeated. Correct?
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Re: If a code word is defined to be a sequence of different [#permalink]

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31 Jan 2015, 07:46
For the life of me,

My calculation does not give me the same answer.

The different part is the 10 Choose 4, my calculations are giving me different figures??

When i factorize top and bottom, i get 252/ 210??

Thanks
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Re: If a code word is defined to be a sequence of different [#permalink]

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31 Jan 2015, 08:10
Tmoni26 wrote:
For the life of me,

My calculation does not give me the same answer.

The different part is the 10 Choose 4, my calculations are giving me different figures??

When i factorize top and bottom, i get 252/ 210??

Thanks

It's not clear what you mean... Anyway, 10C4 = 10!/(6!4!) = 210
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Re: If a code word is defined to be a sequence of different [#permalink]

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31 Jan 2015, 08:18
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
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Re: If a code word is defined to be a sequence of different [#permalink]

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31 Jan 2015, 08:23
Tmoni26 wrote:
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
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Re: If a code word is defined to be a sequence of different [#permalink]

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31 Jan 2015, 08:30
Once again, thanks for your response,

sorry about the notations

My main issue is with the 10 Choose 4 (Denominator) , my understanding is that in order to perform this calculation, I take the first 4 terms of 10! starting with 10 and divide that by 4 factorial.

I do not understand how you get 6! here

Thanks once again and i am very grateful for your time here
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Re: If a code word is defined to be a sequence of different [#permalink]

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31 Jan 2015, 08:32
Tmoni26 wrote:
Once again, thanks for your response,

sorry about the notations

My main issue is with the 10 Choose 4 (Denominator) , my understanding is that in order to perform this calculation, I take the first 4 terms of 10! starting with 10 and divide that by 4 factorial.

I do not understand how you get 6! here

Thanks once again and i am very grateful for your time here

Please follow the link given in my previous post.
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If a code word is defined to be a sequence of different [#permalink]

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31 Jan 2015, 08:33
I am so so sorry, I was doing a Combination instead of a Permutation

Thank you very much
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Re: If a code word is defined to be a sequence of different [#permalink]

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07 Feb 2015, 11:31
In this case ABCDE is different from BCDEA hence order matters boils down to using permutations ...

Hence ( 10! / 5 ! ) * ( 6! / 10 ! ) = 6/1 Answer is E
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If a code word is defined to be a sequence of different [#permalink]

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18 Mar 2016, 02:42
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.
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Re: If a code word is defined to be a sequence of different [#permalink]

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04 Nov 2016, 02:52
Hi, i am confused why we dont care about the order of selection of 5 characters first then 4 characters or 4 characters first then 5 characters. My work is as below, can someone point out where did i do wrong?

First selection:
Ways of 5 characters to be selected in the beginning: 10 x 9 x 8 x 7 x 6 = 30240
Ways of 4 characters to be selected then: 5 x 4 x 3 x 2 =120

Second selection:
Ways of 4 characters to be selected first: 10 x 9 x 8 x 7 = 6860
Ways of 5 characters to be selected then: 6 x 5 x 4 x 3 x 2 = 720

So total of ways for 5 characters: 30960
total of ways for 4 characters: 6980

Ratio 5 characters: 4 characters = 4.436??
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Re: If a code word is defined to be a sequence of different [#permalink]

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05 Jan 2018, 16:46
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

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Re: If a code word is defined to be a sequence of different [#permalink]

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15 May 2018, 08:48
Bunuel wrote:
RadhaKrishnan wrote:
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!
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Re: If a code word is defined to be a sequence of different [#permalink]

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15 May 2018, 09:22
1
panache67 wrote:
Bunuel wrote:
RadhaKrishnan wrote:
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?
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Re: If a code word is defined to be a sequence of different   [#permalink] 15 May 2018, 09:22
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