A plate of cakes contains $12$ different cakes. Determine the number of ways to divide these cakes between Alex and James if each person gets an odd number of cakes.
Another plate has $7$ cup cakes, each decorated with icing of a different colour, and $4$ brownies, each of a different size. Determine how many different line arrangements of these $11$ cakes are possible if no brownie sits beside another brownie.
A plate of biscuits contains $4$ identical chocolate biscuits, $6$ identical shortbread biscuits and $2$ identical gingerbread biscuits. They are all arranged in a row. Determine how many distinct arrangements are possible if the chocolate biscuits stay together.
A plate of biscuits has 4 identical chocolate biscuits, 6 identical shortbread biscuits and 2 identical gingerbread biscuits. All of them are placed in a row. Determine the number of distinct arrangements if the chocolate biscuits are kept as a single group.