14. Diane has five identical blue disks, two identical red disks and one yellow disk. She wants to place them on the grid opposite so that each cell contains exactly one disk. The two red disks are not to be placed in cells that share a common edge.
How many different-looking completed grids can she produce?
A 96
B 108
C 144
D 180
E 216
SOLUTION
B
Note: The key to a problem of this kind is deciding the order in which to consider the placing of the differently coloured disks. In this case it is best to consider first the number of different ways the two red disks may be placed, because they are subject to the condition that they should not be put in cells that share an edge.

In the diagram we have labelled the cells so that we can refer to them.
If the first red disk is placed the cell $P$, then the second red disk may be placed in any one of the 5 cells $R, S, U, V$ and $W$.
Likewise, if it placed in the any of the cells $S, T$ and $Q$, there are 5 possible cells for the second red disk.

If the first red disk is placed any one of the cells $Q, R, U$ and $V$, there are 4 choices for the second red disk.

This gives $5 \times 4+4 \times 4=36$ ways to place the two red disks, but each possible pair of cells has been counted twice.

Therefore there are $36 \div 2=18$ different ways to place the two red disks.
Once the red disks have been placed, there remain 6 cells in which the yellow disk may be placed.
Once the red and yellow disks have been placed, the 5 blue disks must be placed in the remaining 5 empty cells. This may be done in just 1 way.
This gives $18 \times 6 \times 1=108$ different-looking ways of filling the grid.
FOR INVESTIGATION
14.1 How many different-looking grids can Diane produce if she has one yellow disk, three identical red disks and four identical blue disks, and there are no restrictions other than that each cell should contain exactly one disk?
14.2 How many different-looking grids can Diane produce if she has one yellow disk, two identical red disks, two identical blue disks and three identical green disks, and there are no restrictions other than that each cell should contain exactly one disk?
14.3 How many different-looking grids can Diane produce if she has one yellow disk, two identical red disks, two identical blue disks and three identical green disks, with each cell containing exactly one disk and the two red disks not in cells that share a common edge?