##### 4.3
lydiark25
Does anyone understand how to do this problem? I don't get the answer given in the back of the book.
rachelmernoff: Jan. 19, 2015, 8:35 p.m.
3! is the number of ways that the three people can stand next to each other in a line, 5! is the number of ways that the other 5 people can be arranged in a line, and I think that 20 is the number of ways these two groups can be arranged with one another, though I'm not sure exactly how that number is produced.
weisbart: Jan. 20, 2015, 5:16 p.m.
The problem requires three generations of choices---The set in correspondence with the ways the people can line up is a three tuple. The first generation of choice that you make is an ordering of the three people who cannot stand side by side. There are $3!$ orderings. The second generation of choice is a choice of the positions that the three people who cannot stand side by side can occupy. There are 20 such positions. The third generation of choice is a choice of ordering of the remaining five people, $5!$ possibilities. I hope this helps!
mcheng: Jan. 20, 2015, 9:42 p.m.
How do we know that there are 20 positions?
weisbart: Jan. 22, 2015, 9:56 p.m.
You can just count them out, the same way that we did with the frog question. There is an easier way though that we'll talk about tomorrow in class.