RI S1A Permutations and Combinations_tutorial (Solns)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 __________________________________________ Tutorial S1A: Permutations and Combinations Page 1 of 17 Tutorial S1A: Permutations and Combinations Section A (Discussion Questions) 1 9 coloured balls are arranged in a row. Find the number of ways this can be done if (i) all the balls are of different colours, (ii) there are 4 red balls, 3 green balls and 2 yellow balls, (iii) there are 4 red balls, 3 green balls and 2 yellow balls, and the arrangement is symmetrical. [(i) 362880; (ii) 1260; (iii) 12 ] Solution (i) Number of ways = 9! = 362880 9 coloured balls, 4 R, 3 G, 2 Y (ii) Number of ways = 9! 4!3!2! =1260 Alternative method: Out of the 9 positions, choose 4 for the 4 red balls, out of the remaining 5 positions, choose 3 for the 3 green balls : 95 43CC (iii) there are 4 red balls, 3 green balls and 2 yellow balls, and the arrangement is symmetrical. _____ _____ _____ _____ G _____ _____ _____ _____ 2R , 1G, 1Y 2R , 1G, 1Y Since arrangement is symmetrical, middle ball must be green. The first 4 balls must consist 2 red, 1 green and 1 yellow ball. Number of ways = 4! 2! =12 Alternative method: 42 21CC Choose the slots instead.

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ______________________________________ Tutorial S1A: Permutations and Combinations Page 2 of 17 2 How many different three figure numbers between 100 and 999 inclusive, contain two and only two consecutive identical figures? [162] Solution Case 1: First 2 digits are identical. Choices for 1st two digits = 9 (1-9, 0 excluded) Choices for 3rd digit = 9 (any digit from 0-9 that has not been used for 1st digit) Number of ways = 9 x 9 = 81 Case 2: Last 2 digits are identical Choices for 1st digit = 9 (1-9, 0 excluded) Choices for 2nd and 3rd digits = 9 (any digit from 0-9 that has not been used for 1st digit) Number of ways = 9 x 9 = 81 Total number of three figure numbers = 81 + 81 = 162 3 NJC Prelim 9233/2005/01/Q7 (modified) (a) A resort hotel has 3 single rooms, 5 double rooms and 4 family rooms. On a particular week, 2 individuals book a single room each, 3 couples book a double room each and 3 families book a family room each. Given th at all the rooms are available for that week and each room has a distinct design theme, find the number of different possible bookings of rooms amongst the guests. (b) A family of 6 adults and 3 children are to be seated at a round table. Find the number of possible arrangements such that (i) all the 3 children sit together, (ii) none of the children sit together. [ (a) 8640; (b)(i) 4320; (ii) 14400] Solution (a) Rooms :