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Example class 1

Example class 1 - dents three secretaries from fifteen...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 1 1. PGSA will hold a dance competition this semester. Among ten partici- pants, one dancing queen, one first runner-up and one second runner-up are to be selected. Calculate (a) The number of different ways to select the top three; (b) The number of different possible combinations of the participants beyond top three. 2. How many distinguishable outcomes can be generated by tossing five identical tetrahedron simultaneously, if (a) the four sides of each tetrahedron are numbered distinguishably; (b) each tetrahedron has one red side and three blue sides so that sides of the same color are indistinguishable. 3. The Hong Kong University Students’ Union council (HKUSU) will elect a new Executive Committee, including one president, two vice presi-
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Unformatted text preview: dents, three secretaries from fifteen candidates. How many arrange-ments are possible? 1 4. Solution to Question 1. (a) ( 10 3 ) · 3! = 10 × 9 × 8 = 720. Case: Ordered selection without replacement. (b) ( 10 7 ) = 10 × 9 × 8 3 × 2 × 1 = 120. Case: Unordered selection without replacement. 5. Solution to Question 2. (a) r = 5 ,n = 4, ( 4+5-1 5 ) = ( 8 3 ) = 8 × 7 × 6 3 × 2 × 1 = 56. Case: Unordered selection with replacement; x 1 + ... + x 4 = 5, where x 1 ,...,x 4 are non-negative integers. (b) r = 5 ,n = 2, ( 2+5-1 5 ) = ( 6 5 ) = 6. Case: x 1 + x 2 = 5, where x 1 and x 2 are non-negative integers. 6. Solution to Question 3. ( 15 1 , 2 , 3 , 9 ) = 15! 1!2!3!9! = 15 × 14 × 13 × 12 × 11 × 10 2 × 3 × 2 = 300 , 300. Case: Multinomial coefficient. 2...
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