I am having some trouble in questions like selecting $r$ people from $n$ people such that no two people are together. I know that we have to use the GAP method but I don't understand why. I know how to use it while arranging but I don't understand how to use it when we are selecting people.
Eg. We have to select 2 letters from $CHILDREN$ such that they are not together.
When we use gap method we have to place two objects in 7 places. But why?
$\_$ x $\_$ x $\_$ x $\_$ x $\_$ x $\_$ x $\_$
Another doubt that I have is that the two letters that we could place could also be $CH$ like shown:
$C$ x $H$ x $\_$ x $\_$ x $\_$ x $\_$ x $\_$
But in the original word, they are not together. This is another place where using this method seems confusing and wrong to me…
Best Answer
Let's use the gap method for selecting two letters from the word CHILDREN so that the selected letters are not adjacent.
For now, ignore the actual letters. First, we will arrange six blue and two green balls so that the green balls are not adjacent. Place six blue balls in a row. This creates seven spaces, five between successive blue balls and two at the ends of the row.
To ensure that no two green balls are adjacent, we choose two of these spaces in which to insert a green ball. For instance, if we choose the second and fifth spaces above, we obtain the arrangement shown below.
Finally, we label the balls from left to right with the letters of the word CHILDREN. The letters on the green balls are the desired selections of nonadjacent letters from the word CHILDREN. For the example above, we obtain
so the nonadjacent letters we selected were H and R.
Since there are $$\binom{7}{2} = 21$$ ways to choose where the green balls will be placed, there are $21$ ways to select two letters from the word CHILDREN that are not adjacent.
Notice that when we applied the gap method, we ignored the actual letters until after we made the selections. That said, you can verify that the answer is correct by listing all the selections of two nonadjacent letters of the word CHILDREN.