Porisma

Worked solution

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. 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. marks

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

[2011/NJC/II/6]

CJC · Review problems · Question 13

Answer

Committee: 62748. (i) 576; (ii) 1440; (iii) 390624 under the source’s identifiable-diner model.

Choose the committee by its gender composition

The possible compositions are 3 men/5 women,4 men/4 women,5 men/3 women. They contribute (93)(105)=21168\binom93\binom{10}5=21168, (94)(104)=26460\binom94\binom{10}4=26460 and (95)(103)=15120\binom95\binom{10}3=15120. Total 21168+26460+15120=6274821168+26460+15120=62748.

How did this part go?

(i) One women’s block

Four distinct women form one block beside four distinct men: five circular objects, with 4!4! internal women’s orders. Hence (5−1)! 4!=576(5-1)!\,4!=576.

How did this part go?

(ii) Separate the three specified men

Arrange the other five diners circularly in (5−1)!(5-1)! ways. Choose three of their five gaps and arrange Mr Akira, Mr Chan and Mr Ravi in 3!3! ways: (5−1)!(53) 3!=1440(5-1)!\binom53\,3!=1440. Mr Balthus is among the five base diners and has no additional separation restriction.

How did this part go?

(iii) Five choices for each identifiable diner

Each of eight diners has five options: one of four desserts or none. Exclude the one all-none assignment: 58−1=3906245^8-1=390624. This is the source’s diner-by-diner interpretation. The time order of placing the orders does not matter, but which diner chooses which dessert still does. If only aggregate dessert quantities were being counted, it would be a different sample space; the printed answer does not count that model.

How did this part go?

Common mistakes

“Sequence of ordering is not important” removes the order in time, not the identities of the diners in the source model.

Solution notes

All committee cases, all three follow-up asks and working, dessert names, diner-identity assumption and printed marks retained. The source dessert answer is explicitly tied to individual choices rather than aggregate quantities.

Read the hint