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

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28 Jan 2012, 03:59

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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?

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 \(P^4_{10}\);

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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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 -> \(10 * 9 * 8 * 7\)

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

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28 Dec 2012, 06:43

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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: \(=10*9*8*7\)

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 \(P^4_{10}\);

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 \(P^4_{10}\);

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).

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

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16 Apr 2014, 22:24

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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 \(P^4_{10}\);

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?

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

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

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14 Mar 2016, 02:41

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).

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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

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