How many lines of people can be formed?

Question

How many different lines can be formed from three adults and ten children so that the adults are in front and the kids are behind them.I'm not quite sure i solved it right. Any tips? I tried a few to solve this first with.

3*2*1+10*9*8*7*6*5*4*3*2*1

Then with the combination equation.

10!

3!*(10-3)!

= 105

Answer 1

Assumptions: 1) the order of kids and/or adults within a line doesn't matter except that all adults must be before all kids, 2) all adults and kids must be in a line and 3) you must have at least one adult and one kid in a given line.

There are 3 possible combinations of lines with respect to the adults: one line with 3 adults, 2 lines with 2/1 adults and 3 lines with a single adult in each.

The single line has only 1 combination: 3 adults followed by 10 kids.

The pair of lines have 9 combinations, with the line having 2 adults also having 1-9 kids.

The 3 lines have 8 combinations of kids: 8/1/1, 7/2/1, 6/2/2, 6/3/1, 5/3/2, 5/4/1, 4/3/3 and 4/4/2

Now, if the order of the individuals matters, then there are a far greater number of unique combinations. The single-line possibility alone would have 6 combinations of adults and over 3.6M combinations of kids.