arrangements with identical objects.pptx

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1. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Arrangements with “identical” objects 3 rd Quarter

2. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 The Formula The number of arrangements (in a row) of objects when objects are identical, objects are identical, and so on … is: Where  

3. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Example 1 How many possible arrangements of the letters of the word ASSESS are there? Solution: The word ASSESS has 6 letters and thus, we are going to arrange 6 objects in a row. So, . In that word, we have 4 identical objects (the 4 S’s ). Hence, the number of possible arrangements is:  

4. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Example 2 How many possible arrangements of the letters of the word MISSISSIPPI are there? Solution: The word MISSISSIPPI has 11 letters. So, Among the 11 objects, the following are identical: 4 I’s , 4 S’s , & 2P’s . So, the number of possible arrangements is:  

5. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Example 3 An automobile dealer has 3 Fords, 2 Buicks, and 4 Dodges to place in the front row of his car lot. In how many different ways by make of car can he display the automobiles? Solution: The dealer has 9 cars to arrange. Thus, . Assuming that the cars of the same type are identical ( 3 Fords , 2 Buicks , and 4 Dodges ), we have number of possible ways:  

6. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Example 4 In how many ways can we arrange the letters of the word FACTORIZED such that the vowels must appear in the arrangement alphabetically? NOTE: The vowels in the said word are These letters (wherever they are) must appear in the following order: EXAMPLES: DF A C E TR I Z O T AE FDZ I C O R  

7. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Solution We know that we have 4 vowels in consideration. Since they must appear in a specific order, that is, We will disregard (or cancel) any other configurations of these vowels. In other words, we will treat the vowels as if they are identical. So, we have 4 “identical” objects . The word FACTORIZED has 10 letters. So, Thus, the number of possible arrangements is:  

8. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Example 5 In how many ways can we arrange the letters of the word FACTORIZED such that the vowels, and consonants must appear in the arrangement alphabetically? Solution: The vowels in the word are So they must appear in the following order: The consonants in the words are . So they must appear in the following order:  

9. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Solution Since the 4 vowels must appear in a specific order, treat them as if they are “identical”. The logic applies to the 6 consonants . Therefore, the number of possible arrangements is:  

10. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Example 6 Six boys and five girls are to be arranged a line. How many possible arrangements can be made for them if the boys must be put in the line according to their ascending height? Assume that boys have different heights. Solution: Since we are to arrange 6 boys and 5 girls, we have 11 persons to be arranged in a line. So, Now, since the boys (with different heights)must appear in a specific order, we can treat the 6 boys as if they are “identical”. Thus, we have the number of possible arrangements:  

11. SOUTHERN LUZON STATE UNIVERSITY ⃒ LABORATORY SCHOOL [email protected] [phone] / 0949-873-5043 Challenge Problem Take a look again at Example 6. In that problem, we assumed that heights of the boys are different. Consider and answer the following modified problem. Six boys and five girls are to be arranged a line. How many possible arrangements can be made for them if the boys must be put in the line according to their ascending height? In this problem, assume that two of the boys have the same height.