Keep committee choices, circular seating and diner choices distinct

A committee of eight people is chosen from 9 men and 10 women. Find the number of ways it can be chosen with at least 3 men and 3 women.2 marks

In the end four couples—Mr/Mrs Akira, Mr/Mrs Balthus, Mr/Mrs Chan and Mr/Mrs Ravi—form the committee and sit around a table for a meal.

(i) Find the number of arrangements if all the women sit together.2 marks

(ii) Find the number of arrangements if Mr Akira, Mr Chan and Mr Ravi refuse to sit next to each other.2 marks

(iii) Each diner may order at most one dessert from chocolate fudge pie, lemon cake, tiramisu martini and baked Alaska. Find the number of possible selections for the table, given that at least one diner orders a dessert and the sequence of ordering is not important.2 marks