a) Three families celebrate Thanksgiving together: Alvarados, the Bells, and the Carsons. There are four people in each family. Afterwards they line up for a large group photo. How many ways can the 12 of them line up?
b) this time they want to do it so that the Alvarados come first (as you scan from left to right), then the Bells, then the Carsons.Now how many ways can they line up?
c) This time they are going to keep the families together but it doesn't matter what order the families appear (so this time you could have all of the Carsons, followed by all of the Alvarados and then all of the Bells). Now how many ways can they do it?
a) 12! ways = 479, 001, 600 ways
b ) There are 4! ways to arrange each family = 24 so 24 * 24 * 24 = 24^3 = 13824 ways
c) Assuming that each fanily can be arranged in any order....we have 6 ways to order the families * the ways to arrange the people within each family so 6 * 13824 = 82944 ways
Edit to correct my last two answers....
Thank for your help. Can you explain the question (a) How 12! ways = 479, 001, 600 ways? Thank you.
We have 12 ways to choose the first person in line
11 ways to choose the second
10 to choose the third....etc...
12 * 11 * 10 * 9 *.........* 3 * 2 * 1 = 12!