Steven G
Elite Member
- Joined
- Dec 30, 2014
- Messages
- 14,364
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!
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!