Can someone explain the answer but a bit clearer? choose set of 6 diff. letters s.t.

Steven G

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Using an ordered alphabet of 26 letters, how many ways are there to choose a set of six different letters such that no two letters in the set are adjacent in the alphabet?



For instance, {A, C, Q, S, L, Z} is a valid set of six letters, but { A, Q, T, R, Z} is not because Q and R are both in the set.

One way of thinking about this is imagine putting 20 identical objects in a straight line, and then adding 6 objects of another type between them. Now you have a line of 26 objects, 6 of them different from the rest. Apply the alphabet to this line of objects, and imagine that the 6 “distinct” objects highlight 6 letters in the matching position. This is your “answer set”. Notice how the way you place these 6 objects into the rest determines the letters in the answer set, each unique way representing an unique solution. So now the problem becomes “how to place 6 objects between 20 other objects?” The solution is apparent: There are 21 spaces for 6 objects, so the answer is 21C6.

To be honest I am not even clear what is bothering me with this problem. Can someone please simply enlighten me a bit.

Thanks!
 
Using an ordered alphabet of 26 letters, how many ways are there to choose a set of six different letters such that no two letters in the set are adjacent in the alphabet?

For instance, {A, C, Q, S, L, Z} is a valid set of six letters, but { A, Q, T, R, Z} is not because Q and R are both in the set.

One way of thinking about this is imagine putting 20 identical objects in a straight line, and then adding 6 objects of another type between them. Now you have a line of 26 objects, 6 of them different from the rest. Apply the alphabet to this line of objects, and imagine that the 6 “distinct” objects highlight 6 letters in the matching position. This is your “answer set”. Notice how the way you place these 6 objects into the rest determines the letters in the answer set, each unique way representing an unique solution. So now the problem becomes “how to place 6 objects between 20 other objects?” The solution is apparent: There are 21 spaces for 6 objects, so the answer is 21C6.

I would start by imagining what a choice looks like in the context of the whole alphabet. Here is one:

ABCDEFGHIJKLMNOPQRSTUVWXYZ

Since the letters are in fixed positions, this choice can be just as well represented by 26 stars, 6 of which are red:

**************************

Or, to make it easier, 20 stars and 6 bars:

|*|********|****|*|******|

Now, how would we have chosen this, making sure that no two bars are next to one another? We could have started with 20 stars, with (21) single blanks between or around them, so that two bars can't be in the same space:

_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_

Then we chose 6 of those blanks to put bars into:

|*|*_*_*_*_*_*_*_*|*_*_*_*|*|*_*_*_*_*_*|

So there are 21C6 ways to do this.

This is identical to the explanation you were given, but motivates the idea better by starting with the reasoning rather than the answer. Does that work for you?

(Really, the bars could have been any other symbol; stars and bars (or as I call them, sticks and stones) work particularly well for some other problems, so they are familiar.)
 
I would start by imagining what a choice looks like in the context of the whole alphabet. Here is one:

ABCDEFGHIJKLMNOPQRSTUVWXYZ

Since the letters are in fixed positions, this choice can be just as well represented by 26 stars, 6 of which are red:

**************************

Or, to make it easier, 20 stars and 6 bars:

|*|********|****|*|******|

Now, how would we have chosen this, making sure that no two bars are next to one another? We could have started with 20 stars, with (21) single blanks between or around them, so that two bars can't be in the same space:

_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_

Then we chose 6 of those blanks to put bars into:

|*|*_*_*_*_*_*_*_*|*_*_*_*|*|*_*_*_*_*_*|

So there are 21C6 ways to do this.

This is identical to the explanation you were given, but motivates the idea better by starting with the reasoning rather than the answer. Does that work for you?

(Really, the bars could have been any other symbol; stars and bars (or as I call them, sticks and stones) work particularly well for some other problems, so they are familiar.)
You did a great job explaining it as I understand what you are saying perfectly. My biggest concern is that I do not think I could ever have come up with your example/explanation. Of course that is my problem and not yours as you did a great job. I will try to absorb what you said so that I in fact can use this method in the future.

Thank you!

Edit: I thought a bit more about what you did and actually you did not make that big of a leap from other techniques I know. I can and will be able to add this technique (and others!) to my bag of tricks. Probability surely makes you think precisely and carefully. It is a beautiful branch of mathematics.
 
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