(i)[3]
Find how many distinct ways there are to distribute a set of $10$ different mugs between Lucy and Monica, provided that each person receives an odd number of mugs.
(ii)[3]
A different set has $6$ plastic mugs, each with a different design, and $3$ china mugs, each with a different design. Find in how many ways these $9$ mugs can be arranged in a row if the china mugs are all kept apart from one another.
(iii)[3]
A different set contains $3$ identical red mugs, $4$ identical blue mugs and $7$ identical yellow mugs. These $14$ mugs are placed in a row. Find how many different colour arrangements are possible if the red mugs remain together.