How many different ways can a family of 6 sit in a row at the movies if the 2 youngest children must sit next to each other?
First, there are 5 ways the 2 children can be sit. In each case the rest of the family can be sit in 4! different ways. So I think the answer is 5.4!. And it's equal to 120. (Which is 5! So now I see that we could have think the problem as if the 2 children were one person of the family and not 2).
I got a different answer . . .
Duct-tape the two youngest together.How many different ways can a family of 6 sit in a row at the movies
if the 2 youngest children must sit next to each other?
. . Then we have five "people" to arrange: .
And they can be arranged in: . ways.
But in each of those arrangements, can be replaced by
Therefore, there are: . ways.
Let these be the seats
--- --- --- --- --- ---
We first fix the first two seats for the youngest children
[--- ---] --- --- --- ---
There are 4! ways of filling the other 4 seats
Similarly for the cases,
--- [--- ---] --- --- ---
... so on for 5 cases, each of which contributes a 4!. So the total number of ways is 5*4!=120 * 2 = 240 (where 2 children can be swapped)
In general for n seats and k people in connesecutive seats,
[(n-k+1)(n-k)!]k! = (n-k+1)!k!
Same as our answer: 5!.2!=240