annie,betty,Cathy and Danial are going to stand on a row for taking pictures. How many ways can they stand?
4*3*2*1
The first person has the cohoice of 4 spots, takes one
The second person has choice of 3 spots, takes one
The third person has choice of two spots takes one
The fourth person is stuck with the last spot, takes it
4*3*2*1 = 24
permutations of n items taken one at a time.
is n !
n (n-1) (n-2) ....... 1
By the way the permutations of n items taken r at a time is
n!/(n-r)!
and if it does not matter how the r items are arranged in each sub group, then it is the number of "combinations"
n! / [ r! (n-r)! ]
Now what if they were to be paired and we care if each is on left or right in the photo?
that is permutations of four people taken 2 at a time
4!/2! = 4*3*2*1/(2*1) = 12 ways
But what if we do not care if they are on the right or the left. (Betty-Cathy the same as Cathy-Betty)
4!/[2!(4-2)!] = 4*3/2 = 6 ways