Suppose if Ajay and Vijay from the earlier example did not get along and could not picked together. In this case, we can consider three cases - one when Ajay is picked, one when Vijay is picked and one when neither is picked. The sum of these three cases is equal to the number of cases without condition - cases where they are picked together. Number of ways they can be picked together is 715 ways as seen earlier. Hence, number of ways in which at max only one of them is picked = 1365 - 715= 650 ways.
Selection without arrangement:
Choosing r out of n options can be represented as
If the arrangement should always begin or end with a particular item, its order is fixed. Hence the remaining n-1 items can be arranged in (n-1)! ways.
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