Practice section 9 of 10

(a) A teacher wants to set a class test for 25 students. Each student receives the same questions, but in a different order. Find the least number of questions she must set.

(b) Sally has 12 chocolate bars: white 4, milk 3, dark 4 and hazelnut 1. She intends to give one bar each to three of her best friends in school.

(i) Find the number of ways she can do it.

(ii) One particular friend has a nut allergy and should not be given the hazelnut bar. Find the number of ways Sally can now give her friends the chocolates.

(a) Find the number of ways the letters of POSITIVE can be arranged if

(i) all vowels must be next to each other, [2]

(ii) the vowels are separated and the arrangement begins and ends with I. [2]

(b) The Mathematics Department of M Institute consists of 11 teachers, including Mr Tan and Mr Lin. At a meeting, all 11 teachers sit at a round table with numbered seats. How many arrangements have Mr Tan and Mr Lin next to each other? [3]

After the meeting they find a circular table for 6 and a long table with a row of 5 seats at a food court, as shown. Find the number of arrangements of all 11 teachers at these two tables such that Mr Tan is seated next to Mr Lin. [4]

(a) Five couples are to be seated at a round table with numbered seats. Find the number of ways such that each wife must sit next to her husband. [3]

(b) Mr Lin wants to repaint his four bedrooms. Each room can only be painted with one colour. From six different colours, find the number of ways the four rooms can be painted if

(i) there is no restriction on the colours used, [1]

(ii) there is no repetition of colours used, [1]

(iii) he chooses exactly two colours. [3]

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]

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]

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

(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]

Three families comprising six adults and three children go on an outing to a theme park and travel together for the day.

(i) There are 15 amusement rides. A child insists on trying at least 2 of them. Calculate the number of ways this can be done. [2]

(ii) On the Mummy Surprise rollercoaster, a passenger car has seats in 3 rows and 3 columns as shown.

(a) Calculate the number of ways the families can be seated without restrictions. [1]

(b) Tom, the youngest child, insists on sitting next to his mother. Calculate the number of ways this can be done. [2]

(iii) At lunch they are allocated a circular table with 9 seats. Calculate the seating arrangements if none of the children sit together. [2]

Mr and Mrs Tan have a son David and invite 6 guests for dinner. Before dinner, all 9 diners stand in a straight row for a group photo. Find the arrangements if

(i) all 9 diners may stand anywhere, [1]

(ii) the Tan family members must be separated. [2]

The diners then sit at two round tables, a five-seater and a smaller four-seater. Find the arrangements if

(iii) anyone may sit at either table, [2]

(iv) David must sit between his parents at either table. [3]