1. Permutations and Combinations Solutions 2023 (RVHS)

River Valley High School, Mathematics Department 2023 Permutations and Combinations T1 Permutations and Combinations Solutions 1(i) Number of ways ( ) ( ) 3 girls9 people around a circle 6 boys and 1 blo ck of 3 girls Total number of ways without restriction - number of ways all 3 girls together 9 1 ! 7 1 ! 3! 36000 =  = − − −   = Alternative method Number of ways ( ) 3 2 3 girls choose 2 to be in a group6 boys around a circle 6 2 permute w6 slots permute 2 groups of girls number of ways 2 girls together, 1 separated number of ways all 3 girls separated 5 1 ! C P 2! = + − =  ( ) ithin the group of 2 girls 6 3 6 slots permute 3 girls 6 boys around 5 1 ! P 36000      + −  = (ii) Number of ways ( ) 6 3 The 3 girls6 slots choose 36 remaining people around a circle 6 1 ! 3! 14400 C= −   = (iii) Note: The arrangement must look like this Number of ways without restrictions (9 1)! 40320 =− = B B B B B B G G G

River Valley High School, Mathematics Department 2023 Permutations and Combinations T2 Number of ways for exactly 2 boys between any 2 girls ( ) 6 boys permutate within themselvesArranging 3 girls around a circle 3 1 ! 6! 1440 = −  = Required prob. 1440 40320 1 28 = = Alternative Method: Number of ways without restrictions (9 1)! 40320 =− = Number of ways for exactly 2 boys between any 2 girls ( ) ( ) Arranging 3 girls around a circle 36 4 2 2 2 2 6boys choose 2 to slot into first slot 4 boys choose 2 to slot into second slot each group of boys permute within themselves 3 1 ! C C C 2! 1440 = −        = Required prob. 1440 40320 1 28 = = 2(a)(i) Case 1: 2 red + 2 green, no of ways 58 22 280CC=  = Case 2: 1 red + 3 green, no of ways 58 13 280CC=  = Case 3: 4 green, no of ways 8 4 70C== Total no of ways = 280 + 280 + 70 = 630 (ii) No of ways to select at least 1 of each colour = n(any 4 of 13 balls) – n(4 of 5 red) – n(4 of 8 green) = 13 5 8 4 4 4C C C−− = 640 (b) Case 1: blue trousers, i.e. any 2 of 5 skirts: no of ways 5 2 20P== Case 2: green trousers, i.e. no green skirt: no of ways 4 2 12P== Case 3: yellow trousers, i.e. no yellow skirt: no of ways 4 2 12P==

River Valley High School, Mathematics Department 2023 Permutations and Combinations T3 Total no of ways = 20 + 12 + 12 = 44 3 The number of seating arrangements = 4! = 24 Number of ways = 3!  2! = 12 Number of ways = 4!  2!  6 = 288 4(a) No of ways = 10 9 13 840CC= OR 10 9 31 840CC= (i) Form CLC unit: 3 2 2C  ways Seat 8 units in a round table: (8 1)!− ways Total number of ways = 3 2 2 (8 1)!C   − = 30240 (ii) Total number of ways = 30240 10 302400= (b) Case 1: 4 digit number, ending with 2: No of ways = 2  3  2  1 = 12 Case 2: 4 digit number, does not end with 2: No of ways = 2  (2 + 1)  1 = 6 Case 3: 5 digit number, ending with 2: No of ways = 4