MSS Discrete Mathematics [ Matematik Diskrit]

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1 UNIVERSITI SAINS MALAYSIA Pepeiksaan Kusus Semasa Cuti Panjang Sidang Akademik 2011/2012 Ogos 2012 MSS Discete Mathematics [ Matematik Diskit] Duation : 3 hous [Masa : 3 jam] Please check that this examination pape consists of FIVE pages of pinted mateial befoe you begin the examination [Sila pastikan bahawa ketas pepeiksaan ini mengandungi LIMA muka suat yang becetak sebelum anda memulakan pepeiksaan ini] Instuctions: Answe all eight [8] questions [Aahan: Jawab semua lapan [8] soalan] In the event of any discepancies, the English vesion shall be used [Sekianya tedapat sebaang pecanggahan pada soalan pepeiksaan, vesi Bahasa Inggeis hendaklah diguna pakai] 2/

2 - 2 - [MSS 318] (1) Let n and be non-negative integes with n (a) Pove that = n (b) Pove that = n [8 maks] (1) Bia n dan adalah intege tak negatif dengan n (a) Buktikan bahawa = n (b) Buktikan bahawa = n [8 makah] (2) How many pemutations of the lettes ABCDEF G contain (a) the sting BCD? (b) the sting CF GA? (c) the stings BA and GF? (d) the stings ABC and DE? (e) the stings ABC and CDE? (f) the stings CBA and BED? [12 maks] (2) Beapa banyakkah pilihatu bagi huuf-huuf ABCDEF G yang mengandungi (a) entetan BCD? (b) entetan CF GA? (c) entetan BA dan GF? (d) entetan ABC dan DE? (e) entetan ABC dan CDE? (f) entetan CBA dan BED? [12 makah] 3/-

3 - 3 - [MSS 318] (3) Fo the gaphs mentioned in the following statements, give an example o pove that thee is no such gaph: (a) A simple gaph with 6 vetices and 15 edges (b) A simple gaph with 5 vetices, whose degees ae 0, 1, 2, 3, 4, 5 (c) A gaph with 6 vetices that has no Eule cicuit but a Hamiltonian cicuit (d) A simple diected gaph with in-degees 0, 1, 2, 3, 3, 4 and out-degees 1, 1, 2, 2,3,3 [12 maks] (3) Bagi gaf yang dinyatakan dalam penyataan beikut, bei satu contoh atau buktikan tidak tedapat gaf sedemikian: (a) Suatu gaf ingkas dengan 6 bucu dan 15 sisi (b) Suatu gaf ingkas dengan 5 bucu yang mana dajahnya adalah 0, 1, 2, 3, 4, 5 (c) Suatu gaf dengan 6 bucu yang tiada lita Eule tetapi mempunyai lita Hamiltonian (d) Suatu gaf beaah dengan dajah masuk 0, 1, 2, 3, 3, 4 dan dajah kelua 1, 1, 2, 2, 3, 3 [12 makah] (4) Detemine the numbe of solutions fo each of the following (a) x 1 + x 2 + x 3 15, x 1 0, x 2 0, x 3 0 (b) x 1 + x 2 = 10, 0 x 1 4, 0 x 2 5 (c) x 1 + x 2 + x 3 = 10, x 1 5 [12 maks] (4) Tentukan bilangan penyelesaian bagi setiap yang beikut (a) x 1 + x 2 + x 3 15, x 1 0, x 2 0, x 3 0 (b) x 1 + x 2 = 10, 0 x 1 4, 0 x 2 5 (c) x 1 + x 2 + x 3 = 10, x 1 5 [12 makah] 4/-

4 - 4 - [MSS 318] (5) A staight line splits a plane into 2 egions Let R n be the maximum numbe of egions split by n staight lines (a) What ae R 1, R 2, R 3 and R 4? (b) Give a ecuence elation fo R n (c) Find the closed fom fo R n (5) Satu gais luus membahagi satu satah menjadi 2 bahagian Bia R n mewakili bilangan maksimum bahagian yang dipeoleh apabila satu satah tebahagi oleh n gais luus (a) Apakah R 1, R 2, R 3 dan R 4? (b) Beikan hubungan jadi semula bagi R n (c) Dapatkan bentuk tetutup bagi R n [14 makah] (6) You have won a pice! Now, you have 2 options The fist option is to get RM1 million in each of the following 100 days The second option is to get RM1 in the fist day Then 20% moe than the pevious day in each of the following 99 days That means you will get RM120 fo the second day, RM144 fo the thid day, and so on Which one would you pefe? Justify you answe (6) Anda telah memenangi suatu hadiah! Sekaang, anda mempunyai 2 pilihan Pilihan petama adalah anda mendapat RM1 juta sehai bagi 100 hai Pilihan kedua adalah anda mendapat RM 1 untuk hai petama Kemudian, tambahan 20% lebih daipada hai yang sebelumnya selama 99 hai Ini bemaksud anda akan mendapat RM 120 untuk hai kedua, RM 144 untuk hai ketiga, dan seteusnya Yang mana satukah pilihan anda? Tentusahkan jawapan anda [14 makah] (7) (a) Find the chomatic numbe of the following gaphs (i) K 6 (ii) K 5,4 (iii) C 5 (b) Conside the bipatite gaph K 3,4 (i) Daw the bipatite gaph K 3,4 (ii) Find a 2-coloing of the gaph (iii) Find all the cycles of a vetex and show that they ae even 5/-

5 - 5 - [MSS 318] (7) (a) Dapatkan nombo komatik bagi gaf-gaf yang beikut (i) K 6 (ii) K 5,4 (iii) C 5 (b) Petimbangkan gaf dwi patisi K 3,4 (i) Lukis gaf dwi patisi K 3,4 (ii) Dapatkan 2-pewanaan bagi gaf tesebut (iii) Dapatkan kesemua kita bagi suatu bucu dan tunjukkan meeka adalah genap [14 makah] (8) (a) Using geneating functions, detemine the numbe of ways nine identical books can be distibuted among thee distinct childen if each child eceives at least two books and no moe than fou books (b) Use geneating functions to find the numbe of ways to select objects of n diffeent kinds if we must select at least one object of each kind (8) (a) Menggunakan fungsi penjana, tentukan bilangan caa sembilan buku yang sama boleh diagihkan kepada tiga kanak-kanak yang bebeza jika setiap kanakkanak meneima sekuang-kuangnya dua buah buku dan tidak lebih daipada empat buah buku (b) Gunakan fungsi penjana untuk mendapatkan bilangan caa memilih objek daipada n jenis bebeza jika kita haus memilih sekuang-kuangnya satu objek daipada satu jenis [14 makah] - ooo O ooo -

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