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ruis
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­In a law firm, each case is to be designated by a three-letter-code 02 Mar 2024, 05:09
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25% (01:59) correct 75% (01:34) wrong based on 101 sessions

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­In a law firm, each case is to be designated by a three-letter-code in which no letter can be used more than twice. If there are 26 letters to choose from, how many different codes are available? Two codes with the same letters in a different order are considered different codes. 

A. 26^3 
B. 27*26*25
C. 26^2 *25
D. 26*25*3
E. 26*25*24­­
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­In a law firm, each case is to be designated by a three-letter-code 02 Mar 2024, 06:34
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ruis
­In a law firm, each case is to be designated by a three-letter-code in which no letter can be used more than twice. If there are 26 letters to choose from, how many different codes are available? Two codes with the same letters in a different order are considered different codes. 

A. 26^3 
B. 27*26*25
C. 26^2 *25
D. 26*25*3
E. 26*25*24­­
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Let's represent the three-digit code as - 
_ _ _

Without any restriction, the above places can be filled as follows  -
  • The first place can be filled in 26 ways
  • The second place can be filled in 26 ways
  • The third place can be filled in 26 ways

Number of ways of forming a code with 26 available letters = \(26^3 \)ways

Restriction Mentioned in the question:  No letter can be used more than twice.

If a letter were used twice, the code would have all three letters the same.

Number of ways of forming a code with 26 available letters such that all letters are the same = 26 ways

Number of ways of forming a code with 26 available letters such that no letter can be used twice = \(26^3 - 26 = 26*(26^2 - 1) \)
= 26 (26-1)(26+1)
= 26 * 25 * 27

Option B


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­In a law firm, each case is to be designated by a three-letter-code 02 Mar 2024, 06:52
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ruis
­In a law firm, each case is to be designated by a three-letter-code in which no letter can be used more than twice. If there are 26 letters to choose from, how many different codes are available? Two codes with the same letters in a different order are considered different codes. 

A. 26^3 
B. 27*26*25
C. 26^2 *25
D. 26*25*3
E. 26*25*24­­
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Different arrangements gives us different codes. Each letter can be used multiple times (for now, let's ignore the no more than twice restriction).
In this case, we will use the fundamental counting principle.

So the number of codes will simply be = 26 * 26 * 26

The only limitation is that a letter cannot be used thrice i.e. AAA, GGG etc are not acceptable. How many such cases are there? Only 26 because there are 26 letters. 

Hence acceptable codes \(= 26^3 - 26 = 26 (26^2 - 1) = 26 * (26+1)(26 -1) = 27*26*25\)

Answer (B)

Check fundamental counting principle here in this video. 

­
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­In a law firm, each case is to be designated by a three-letter-code 04 May 2024, 12:33
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There can be two options of making a code.
FIRST - All three letters are different.
For example- A B C, then total ways of making code will be - 26 (first letter) * 25 (second letter) * 24 (third letter)

SECOND - Two letters gets repeated and third one is different.
For example- A A C, then total ways of making code will be - 26 (first letter for two slots) * 25 (second letter) * 3!/2! (Total different ways to adjust 3 letters having two identical letters- eg. A A C)

Add FIRST & SECOND
=26*25*24 + 26*25*3!/2!
=26*25 [24+3]
=26*25*27
That's Option B
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­In a law firm, each case is to be designated by a three-letter-code 05 May 2024, 22:44
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Can someone explain the reasoning behind why 26*26*25 would not work in this case? Since for the first option, we have 26 choices. For the second option. We have 26 choices again but for the third option, since we can’t repeat a letter more than twice, we only have 25 options?

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­In a law firm, each case is to be designated by a three-letter-code 06 May 2024, 00:57
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­In a law firm, each case is to be designated by a three-letter-code in which no letter can be used more than twice. If there are 26 letters to choose from, how many different codes are available? Two codes with the same letters in a different order are considered different codes. 

A. 26^3 
B. 27*26*25
C. 26^2 *25
D. 26*25*3
E. 26*25*24­

Can someone explain the reasoning behind why 26*26*25 would not work in this case? Since for the first option, we have 26 choices. For the second option. We have 26 choices again but for the third option, since we can’t repeat a letter more than twice, we only have 25 options?

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There can be two cases:

XXY, two of the three letters are the same, and XYZ, all three letters are different:

For XXY case X's can take 26 options, and Y can take 25. However, since we can have XXY, XYX, or YXX sequences, the total for this case would be 26*25*3.

For XYZ case, we'd have 26*25*24 possibilities. Note here that this number already accounts for all the permutations of XYZ. Alternatively, we could do choosing 3 out of 26 when the order of the selection matters, which is given by 26P3 = 26!/(26 - 3)! = 26*25*24.

Thus, total would be 26*25*3 + 26*25*24 = 26*25(24 + 3) = 27*26*25.

Answer: B­
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­In a law firm, each case is to be designated by a three-letter-code 03 Jun 2025, 23:51
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No letter can be used more than twice => a letter can be used maximum 2 times in the 3-letter code.

So, here are the allowed cases for the 3-letter code ->

(1) All 3 letters different -> example "abp"

26C3 ways of choosing the 3 letters and 3! ways to arrange the 3 chosen letters

i.e., 26C3 x 3! = 26 x 25 x 24


(2) One letter is used twice -> example "aab" or "aca" or "xab"

26C2 ways of choosing the 2 letters and 2C1 ways of then choosing the letter that will repeat from among the 2 chosen letters and 3!/2! ways of then arranging the chosen letters (x,x,y -> 3!/2! arrangements)

i.e., 26C2 x 2C1 x \(\frac{3!}{2!}\) = 26 x 25 x 3


Final answer: (1) + (2) = 26 x 25 x (24+3) = 27 x 26 x 25. Choice B.

Harsha
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