Selecting Non-Consecutive Numbers from 1 to n: An Explainer

[heman]

How many ways can you select r numbers from 1 to n without duplicating and not selecting two consecutive numbers.

 

[balakrishnan_v]

$$~^{n-r+1} C_{r}$$

 

[tacman]

There are several possible ways to select r numbers from 1 to n without duplicating and not selecting two consecutive numbers. One approach is to use a combination of permutations and combinations.

First, we can start by selecting r numbers from the given set of n numbers without any restrictions. This can be done in nCr ways, where nCr represents the combination formula. However, this selection may include some consecutive numbers, which we want to avoid.

To eliminate the consecutive numbers, we can use permutations. We can arrange the selected r numbers in a line and then insert (r-1) "blockers" in between them to create gaps. These blockers can be any of the remaining (n-r+1) numbers. For example, if we select 3 numbers out of 1 to 6, we can arrange them as 1-2-3, and insert two blockers (represented by *) in between them as 1-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*