There are 20 people on a committee. In how many ways can they be lined up if Rio and Lima will NOT stand together?
To find the number of ways the 20 people can be lined up such that Rio and Lima do not stand together, we can use the principle of complementary counting.
We first count the total number of ways to line up the 20 people without any restrictions. In this case, all 20 people can be arranged in a line in 20! (20 factorial) ways.
Next, we calculate the number of ways that Rio and Lima stand together. Here, we can treat Rio and Lima as a single entity, which means they can be arranged among themselves in 2! (2 factorial) ways. The remaining 19 people (excluding Rio and Lima) can be arranged in 19! ways.
So, the number of lineups where Rio and Lima stand together is 2! * 19!.
Finally, we subtract the number of lineups where Rio and Lima stand together from the total number of lineups to get the desired result:
Total number of lineups - Number of lineups where Rio and Lima stand together
= 20! - (2! * 19!)
Now we can calculate the answer:
20! - (2! * 19!)
= 20! - (2 * 19!)
= 20 * 19! - (2 * 19!)
= 19!(20 - 2)
= 18 * 19!
Therefore, there are 18 * 19! ways in which the 20 people can be lined up if Rio and Lima will not stand together.