Worksheet: Permutations and Combinations (solutions)
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This document contains solutions to probability theory problems involving permutations and combinations. It solves four problems: 1) finding the number of ways to fill four teaching positions from 16 applicants with and without considering order; 2) finding the number of anagrams of MASSACHUSETTS; 3) finding the number of ways to get different scores on a true/false test; and 4) reworking the MASSACHUSETTS anagram problem by choosing slots for each letter. The solutions use permutations and combinations to systematically calculate the number of possible outcomes for each scenario.
Worksheet: Permutations and Combinations (solutions)
- 1. V63.0233: Theory of Probability Solutions Worksheet for Sections 1.3–1.4 : Permutations and Combinations June 30, 2009 1. Among the 16 applicants for four different teaching positions in an elementary school, only ten have master’s degrees. (i) In how many ways can these positions be filled? (ii) In how many ways can these positions be filled with applicants having master’s degrees? (iii) If one of the positions requires a master’s degree, while for the others it’s optional, in how many ways can the four positions be filled? Solution. (i) Notice that the positions are described as different so order matters. We have 16 ways to fill the first spot, 15 to fill the second, etc., until you fill four spots: 16 × 15 × 14 × 13 = 43, 680 (ii) We have 10 ways to fill the first, spot, 9 ways to fill the second, and so on: 10 × 9 × 8 × 7 = 5040 (iii) 2. How many anagrams of the word MASSACHUSETTS can you find? Solution. Of the 13 letters, we have two A’s, four S’s, and two T’s. This gives a total of 13! = 64, 864, 800 4!2!2! anagrams. 1
- 2. 3. A student takes a true-false test of 15 questions. In how many different ways can he or she mark this test and get (i) three right and 12 wrong? (ii) six right and nine wrong? (iii) 12 right and three wrong? Solution. (i) We can choose the incorrect three to be any of the questions numbered 1 through 15. Hence there are ( ) 15 = 455 3 different ways to do this. (ii) ( ) 15 = 5005 6 (iii) ( ) 15 = 455 12 (Notice there are exactly the same number of ways to get three right as there are to get three wrong.) 4. Rework the MASSACHUSETTS problem by choosing the slots for each of the letters. For instance, there are 13 positions for the M. Of the 12 remaining, we need to choose two for the Ts. Of the 10 remaining, we need to choose four for the Ss, etc. Solution. Following the directions, we can choose ) of 13 positions for the M. The 2 As need to ( any 12 go in any of the remaining 12 slots; there are ways to do this. And so on. We get 2 ( )( )( )( )( )( )( )( ) 13 12 10 6 5 4 3 2 = 64, 864, 800 1 2 4 1 1 1 1 2 M A S C H U E T This is the same thing we get in Problem 2. 2