One of the solved examples in the book A First Course in Probability by Sheldon M. Ross goes as follows:
There are 20 people made up of 10 married couples. Choose 5 people from this group of 20 such that no two are married to each other.
I worked out the problem before looking at the solution provided and found out to my dismay that I got a very fundamental aspect wrong.
My approach was as follows: Sample Space = Choose 5 people from the group of 20.
$\therefore S$ = $20\choose 5 \\$
Now we count the number of ways unmarried people get chosen.
- For the first person we have 20 choices.
- For the next person we have 18 choices (excluding 1st person and his spouse)..... and so on.
So number of events = $20 \times 18 \times 16 \times 14 \times 12$
Probability = $\frac{20 \times 18 \times 16 \times 14 \times 12}{20\choose 5}$
However this is not correct.
According to the book, either the sample space calculation has to be changed to ordered selection, $20 \times 19 \times 18 \times 17 \times 16 \\$ or, the number of events has to be changed to unordered selection $10\choose 5$ $\times$ $2^5$.
I understand how each of these term were calculated individually. What I am not a hundred percent clear on is: Why can't ordered selection be mixed with unordered selection? Exactly where is my sample space calculation not compatible with the calculation of number of events?
Just as somebody has already pointed out above it makes no sense to mix up ordered and unordered choice. To make your solution work, you need to multiply the denominator by 5! (5 factorial - to not mix up the meaning of the exclamation mark (:) (to have all possible permutations of those 5 chosen people).
Ordered choice=Unordered choice*The number of chosen things (factorial)
Then the result is the same as the ones you’ve written.
Have a nice Sunday!