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Each runner during a race is labeled with a unique [#permalink]

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16 Sep 2004, 19:54

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Each runner during a race is labeled with a unique one-letter code or a unique two-letter code, where the two letters are different. The codes use the 26-letter English alphabet. What is the maximum number of runners that can receive unique codes for the race?

676
26 + 26^2 - 26 = 26^2 = 676
First 26 is just the 26 individual letters
26^2 is for all possible combinations of 2 alphabet letters
Last 26 is unfavorable outcomes when 2 letters are similar ie AA,BB,CC... There are 26 of these possibilities
_________________

Each runner during a race is labeled with a unique one-letter code or a unique two-letter code, where the two letters are different. The codes use the 26-letter English alphabet. What is the maximum number of runners that can receive unique codes for the race?

26 unique one-letter codes
(26*25) unique two letter codes where two letters are different = 650

total no. of alphabets = 26 pattern of code: 12 At place 1 we can have null or 26 characters. At place 2 we can have 26 characters or 25 characters(depending if null was at 1 or not) total becomes 1*26 + 26*25 = 26+ 650 = 676

or faster way is to start from right side at place 2, we have 26 characters at place 1 we have 25 character +1 null= 26 26*26=676

Unique letter combinations are 26 two letter combinations are = 26c2 = 26X25/2 = 13X25 = 325 two letter code where two letters are same = 26

total combinations = 26+325-26 = 325

answer should be 325

Would you say AC and CA are different codes or same? I think you will agree that they are different codes. Hence you do not use the combination formula here (26C2) because it only selects 2 different letters out of the 26 letters.

So how do you solve this question? You can do it in two different ways:

Method 1: You select 2 different letters and then arrange them. 26C2 * 2! = 650

Unique letter combinations are 26 two letter combinations are = 26c2 = 26X25/2 = 13X25 = 325 two letter code where two letters are same = 26

total combinations = 26+325-26 = 325

answer should be 325

Would you say AC and CA are different codes or same? I think you will agree that they are different codes. Hence you do not use the combination formula here (26C2) because it only selects 2 different letters out of the 26 letters.

So how do you solve this question? You can do it in two different ways:

Method 1: You select 2 different letters and then arrange them. 26C2 * 2! = 650

The number of one letter codes is 26 so total = 650 + 26 = 676

I agree with you and understood your point. however in one of kaplan CAT the OA is porvided as 351. Please see the attachment. Please anyone explain why OA is 351

Unique letter combinations are 26 two letter combinations are = 26c2 = 26X25/2 = 13X25 = 325 two letter code where two letters are same = 26

total combinations = 26+325-26 = 325

answer should be 325

Would you say AC and CA are different codes or same? I think you will agree that they are different codes. Hence you do not use the combination formula here (26C2) because it only selects 2 different letters out of the 26 letters.

So how do you solve this question? You can do it in two different ways:

Method 1: You select 2 different letters and then arrange them. 26C2 * 2! = 650

The number of one letter codes is 26 so total = 650 + 26 = 676

I agree with you and understood your point. however in one of kaplan CAT the OA is porvided as 351. Please see the attachment. Please anyone explain why OA is 351

Attachment:

Kaplan CAT.PNG

Yes, because the question in the attachment is different from the main post. It states an additional condition that if a 2 letter code is used, the reverse cannot be used.

So, AB- Is one valid code BA- will not be a valid code because AB has already been used.

Whereas according to the main question, both AB and BA will be considered valid codes.

Thanks fluke. This is the precise reason I opened a saparate topic as combinatorics-kaplan-700-level-ps-unique-2-latter-code-122067.html. This was locked by the 144144. can you please open the thread so that discussion can go forward
_________________

Each runner during a race is labeled with a unique one-letter code or a unique two-letter code, where the two letters are different. The codes use the 26-letter English alphabet. What is the maximum number of runners that can receive unique codes for the race?

Ans- 676

Use slot method to solve this question:

For unique one letter code: 26 For unique two letter code: 26*25 = 650

Total # of codes = 26 + 650 = 676
_________________

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