Module 8: Lesson 2

 

Self-Check 7

 

“Check Your Understanding” questions 8.b., 8.c., 17, and 21 on pages 525 and 526 of the textbook

8.  
   2. Each unit that is travelled to the right can be represented by the letter R and each unit that is travelled up can be represented by the letter U. Using these letters, one possible pathway is RRRRUUU. The “word” RRRRUUU can be treated as though it were a seven-letter word with four repetitions of the letter R and three repetitions of the letter U. The number of different paths is
      
      
       
      
   3. Each unit that is travelled up can be represented by the letter U and each unit that is travelled to the left can be represented by the letter L. Using these letters, one possible pathway is UULLL. The “word” UULLL can be treated as though it were a five-letter word with two repetitions of the letter U and three repetitions of the letter L. The number of different paths is
      
      
       
      

17.  
    1. There are eight letters in parallel. There are 8! ways to arrange these eight letters. Of the eight  letters, however, three are the letter l and two are the letter a. There are 3! ways to arrange the letter l and 2! ways to arrange the letter a. The number of different eight-letter arrangements is
       
       
        
       
    2. There is more than one way to solve this question. Only one method follows.
       
       Three of the eight letters in parallel are the letter l. They can be arranged in 3! ways. The other five letters can be arranged in 5! ways. The three l’s do not have to be placed at the beginning of the word, however. There are six different positions where the three l’s can be placed and still be together. Thus, there are 3! x 5! x 6 ways in which the letters can be arranged and have all of the l’s together. Some of these ways are repetitions of others, however, so these need to be removed:
       
       
        
       
       
       There are 360 different arrangements in which all of the l’s would be together.

21.  

    1. Each sonnet is 14 lines and each line has 10 different possibilities, so the number of possible arrangements is 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 or 1014 possible arrangements.

    2. The number 1014 when expanded is 100 000 000 000 000. That number is 100 million million, so the title of the book is reasonable.

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