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How many 2-letter "words" consist of two different letters arranged in alphabetical order? (Any two letters together is considered a "word." For example, one such word is DQ.)

rohitaop  May 18, 2018
 #1
avatar+797 
+1

Hey rohitaop!

 

Here is my solution:

 

We can consider the cases:

 

If A is the first letter, we have B-Z possibilities, or 25 letters. 

 

If B is the first letter, we have C-Z possibilities, or 24 letters. 

.

.

.

If Y is the first letter, we have Z possibilities, or 1 letter. 

 

We have the sum:

 

1 + 2 + 3 + 4 ... + 24 + 25, we use the arithmetic sum formula where the difference is 1. 

 

25 * (25+1) / 2 = 325.

 

I hope this helped,

 

Gavin

GYanggg  May 18, 2018
 #2
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0

What is so "special" about these combinations? They are simple binomials:

26C2 = 325, in alphabetical order: AB, AC, AD.........etc.

Guest May 18, 2018
 #3
avatar+1048 
0

No! Mr. BB., they are not simple binomials in alphabetical order. Gobble! Gobble!

GingerAle  May 18, 2018
 #4
avatar+1048 
+1

This is not the correct solution for this question.

For combinations, a set such as AB = BA, and the two sets are the same and counted only once. If this were the case, then 26C2 would be the correct solution. \(26C2 = \dfrac {26!}{2!(26-2)!} = 325\)

 

In this question, the order of the sets is unique and counting the permutations of these sets gives the correct solution.

\(26P2 = \frac {26!}{(26n-2)!} = 650\)

Note that the formulas differ only by the division of the set size factorial (2!).

 

Here’s another method for solving.

Counting all combinations gives \(26^2 = 676\) sets of two letters including AA , BB, ect.

The double letters occurs once for each of the 26 letters. Subtracting the 26 duplicates from 676 leaves 650 sets, and this the number of sets without a repeated letter.

 

Note also that alphabetizing does not make a difference in the set counts. 

 

 

GA

GingerAle  May 18, 2018
edited by GingerAle  May 18, 2018
 #5
avatar
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How many 2-letter "words" consist of two different letters arranged in alphabetical order?

 

 

the guest that you were so eager to criticize was right.

Guest May 21, 2018

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