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Unformatted text preview: Solution to HW 4 Ning SUN August 11, 2009 5.3: 2, 22; 5.4: 2, 12, 34, 48; 6.1: 2 (b,e), 4 (b, d), 8; 6.2: 2, 8, 18(a), 20 5.3#2 MISSISSIPPI: 1M, 4I, 4S, 2P, to order them: arrangement + restricted repetition, use the formula P ( n ; r 1 ,...,r n ) = P (11; 1 , 4 , 2 , 4) = 11! 4!2!4! = 34 , 650 . 5.3#22 PEPPERMILL: 3P, 2E, 2L,1M,1I, 1R 1. M to the left of all the vowels (2E and I): Find the positions for the the subset (M,E,E,I) ( C (10 , 4) ways), order them and put the M in the rst position of the subset (3 ways: MEEI, MIEE, MEIE). Then order the remaining 6 letters: 3P, 2L, 1R ( P (6; 3 , 2 , 1) ways). So in total, there are C (10 , 4) · 3 · P (6; 3 , 2 , 1) = 37 , 800 ways. 2. The rst P appears before the rst L: Find the positions for the subset (P,P,P,L,L) ( C (10 , 5) ways), order them and put a P in the rst position of the subset (2P and 2L in the remaining 4 positions of the subset, so P (4; 2 , 2) =6 ways). Then order the remaining 5 letters: 2E, 1M,1I and 1R ( P (5; 2 , 1 , 1 , 1) = 20 ). So in total, there are C (10 , 5) · 6 · 20 = 30 , 240 ways. 5.4#2 1. arrangement + unlimited repetition, use the formula n r = 4 16 = 4 , 294 , 967 , 296 . 2. arrangement+ restricted repetition, use the formula P ( n ; r 1 ,...,r n ) = P (16; 6 , 6 , 2 , 2) = 16! 6!6!2!2! = 10 , 090 , 080 . 3. arrangement+ restricted repetition, use the formula P ( n ; r 1 ,...,r n ) = P (16; 4 , 4 , 4 , 4) = 16!...
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