RI 2022 H3 Test 3(Questions and Solutions)

RI 2022 RAFFLES INSTITUTION 2022 Year 6 H3 9820 Lecture Test 3 Time allocated: 2 hours Total Marks: 60 Instructions: Write your name and CT group on all the work you hand in. Answer all questions. 1 A clothes shop sells a particular make of T-shirt in four different colours. The shopkeeper has a large number of T-shirts of each colour. (a) A customer wishes to buy eight T-shirts. (i) In how many ways can he do this? [2] (ii) In how many ways can he do this if he buys at least one of each colour? [2] (b) The shopkeeper places eight T-shirts in a line. (i) In how many ways can she do this? [1] (ii) In how many ways can she do this if no two T-shirts of the same colour are to be next to each other? [2] (iii) Use the principle of inclusion and exclusion to find the number of ways in which she can do this if she has to use at least one T -shirt of each colour but with no other restriction. [4] (c) The shopkeeper is left with r T-shirts of one colour, where 4r and another 2 T-shirts of another colour. She wishes to store them in 2 identical boxes so that no box is empty. In how many ways can she do this? [4] 1 (a) (i) [2] Let 1 2 3 4, , ,x x x x be the respective number of T -shirts the customer buys in the 4 different colours. Then the problem is equivalent to the number of integer solutions to 1 2 3 4 8x x x x+ + + = , with 1 2 3 4, , , 0x x x x  . (Identical objects into distinct boxes) The number of ways is thus 11 165.3  =

RI 2022 (a)(ii) [2] Let 1 2 3 4, , ,x x x x be the respective number of T -shirts the customer buys in the 4 different colours. Then the problem is equivalent to 1 2 3 4 8x x x x+ + + = , with 1 2 3 4, , , 1x x x x  . Let 1iiyx=− (i.e. ensure he has one of each different col our). Then we need to find the number of integer solutions to 1 2 3 4 4y y y y+ + + = with 1 2 3 4, , , 0y y y y  . The number of ways is thus 7 353  = . (b)(i) [1] The number of ways is 84 65536= . (b)(ii) [2] There are 4 ways to choose the first shirt, and subsequently, 3 ways for the next shirt so that no two shirts are of the same colour. The number of ways is thus 74(3) 8748= . (b)(iii) [4] Let iA denote the event in each colour i is not used. Then required answer = __ __ __ __ 8 1 2 3 4 1 2 3 4 4A A A A A A A A   = −    We have 1 2 3 4 1 2 3 4A A A A A A A A   = + + + 1 2 1 3 3 4A A A A A A−  −  − −  1 2 3 1 2 4 2 3 4A A A A A A A A A+   +   +   1 2 3 4A A A A−    ( ) 8884 4(3) 6 2 4 40824= − + − = .

RI 2022 (c) [4] Consider the 2 T-sh