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A large pile of coins consists of pennies, nickels, dimes, and quarters. Use the method shown in Example 9.6.2 to answer the following questions. (a) How many different collections of 40 coins can be chosen if there are at least 40 of each kind of coin? 12341 collections (b) If the pile contains only 20 quarters but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? X collections (c) If the pile contains only 30 dimes but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? collections

A large pile of coins consists of pennies, nickels, dimes, and quarters. Use the method shown in Example 9.6.2 to answer the following questions. (a) How many different collections of 40 coins can be chosen if there are at least 40 of each kind of coin? 12341 collections (b) If the pile contains only 20 quarters but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? X collections (c) If the pile contains only 30 dimes but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? collections

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Example 9.6.2 Selecting 15 Cans of Soft Drinks of Five Different Types A person giving a party wants to set out 15 assorted cans of soft drinks for his guests. He shops at a store that sells five different types of soft drinks. a. How many different selections of cans of 15 soft drinks can he make? b. If root beer is one of the types of soft drink, how many different selections include at least six cans of root beer? C. If the store has only five cans of root beer but at least 15 cans of each other type of soft drink, how many different selections are there? ors nod 2wank or Solution a. Think of the five different types of soft drinks as the n categories and the 15 cans of soft drinks to be chosen as ther objects (so n = 5 and r = 15). Each selection of cans of soft drinks is represented by a string of 5-1 = 4 vertical bars (to separate the categories of soft drinks) and 15 crosses (to represent the cans selected). For instance, the string odmun odi oos x x × | × X x x × × ||× × × | × X represents a selection of three cans of soft drinks of type 1, seven of type 2, none of ) ted Isor type 3, three of type 4, and two of type 5. The total number of selections of 15 cans of to odo of soft drinks of the five types is the number of strings of 19 symbols, 5-1= 4 of them| ed and 15 of them X: to 1odr (*)- 6 2 19.18 17 16 15! 15+5-1 19! (19) 15 151 -4-3-2-1 = 3,876. 15!·(19 – 15)! 15 b. If at least six cans of root beer are to be included in the selection, you can imagine choosing six such cans first and then choosing nine additional cans. The choice of the nine additional cans can be represented as a string of 9 ×'s and 4 's. For example, if root beer is type 1, then the string X × X || X X |X X X X | represents a selection of three cans of root beer (in addition to the six chosen initially), none of type 2. two of type 3, four of type 4, and none of type 5. Thus the total number of selections of 15 cans of soft drinks of the five types, including at least six cans of root beer, is the number of strings of 13 symbols, 4 (=5 – 1) of them | and 9 of them ×: 9+4 13 12-11 10-9! 91 -4-3-2-1 13! 13 9!-(13 – 9)! 9 = 715.
Transcribed Image Text:Example 9.6.2 Selecting 15 Cans of Soft Drinks of Five Different Types A person giving a party wants to set out 15 assorted cans of soft drinks for his guests. He shops at a store that sells five different types of soft drinks. a. How many different selections of cans of 15 soft drinks can he make? b. If root beer is one of the types of soft drink, how many different selections include at least six cans of root beer? C. If the store has only five cans of root beer but at least 15 cans of each other type of soft drink, how many different selections are there? ors nod 2wank or Solution a. Think of the five different types of soft drinks as the n categories and the 15 cans of soft drinks to be chosen as ther objects (so n = 5 and r = 15). Each selection of cans of soft drinks is represented by a string of 5-1 = 4 vertical bars (to separate the categories of soft drinks) and 15 crosses (to represent the cans selected). For instance, the string odmun odi oos x x × | × X x x × × ||× × × | × X represents a selection of three cans of soft drinks of type 1, seven of type 2, none of ) ted Isor type 3, three of type 4, and two of type 5. The total number of selections of 15 cans of to odo of soft drinks of the five types is the number of strings of 19 symbols, 5-1= 4 of them| ed and 15 of them X: to 1odr (*)- 6 2 19.18 17 16 15! 15+5-1 19! (19) 15 151 -4-3-2-1 = 3,876. 15!·(19 – 15)! 15 b. If at least six cans of root beer are to be included in the selection, you can imagine choosing six such cans first and then choosing nine additional cans. The choice of the nine additional cans can be represented as a string of 9 ×'s and 4 's. For example, if root beer is type 1, then the string X × X || X X |X X X X | represents a selection of three cans of root beer (in addition to the six chosen initially), none of type 2. two of type 3, four of type 4, and none of type 5. Thus the total number of selections of 15 cans of soft drinks of the five types, including at least six cans of root beer, is the number of strings of 13 symbols, 4 (=5 – 1) of them | and 9 of them ×: 9+4 13 12-11 10-9! 91 -4-3-2-1 13! 13 9!-(13 – 9)! 9 = 715.
A large pile of coins consists of pennies, nickels, dimes, and quarters. Use the method shown in Example 9.6.2 to answer the following questions. (a) How many different collections of 40 coins can be chosen if there are at least 40 of each kind of coin? 12341 collections (b) If the pile contains only 20 quarters but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? X collections (c) If the pile contains only 30 dimes but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? collections (d) If the pile contains only 20 quarters and only 30 dimes but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? collections
Transcribed Image Text:A large pile of coins consists of pennies, nickels, dimes, and quarters. Use the method shown in Example 9.6.2 to answer the following questions. (a) How many different collections of 40 coins can be chosen if there are at least 40 of each kind of coin? 12341 collections (b) If the pile contains only 20 quarters but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? X collections (c) If the pile contains only 30 dimes but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? collections (d) If the pile contains only 20 quarters and only 30 dimes but at least 40 of each other kind of coin, how many collections of 40 coins can be chosen? collections
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