David W. answered • 07/29/17
Tutor
4.7
(90)
Experienced Prof
First, note that there are two concerns with the wording of this problem:
(1) "One-thousand dollar bills may seem rare and exotic to the casual observers. The latest series date is 1934A. Surprisingly though, $1,000 bills were seen in circulation up until the early 1970s. You could still order them from banks in the 1960s. It is not uncommon for older Americans to take them to the bank to try to get face value today. Of course, we wouldn’t recommend selling them for face value. Only severely impaired examples should be worth face value. All other notes are definitely collectible. Some $1,000 bills can be worth several thousand dollars each. Your standard value for a generic note in lightly circulated condition is probably $1,600. However, there are plenty of exceptions to that rule." See --http://www.antiquemoney.com/old-one-thousand-dollar-bill-value-price-guide/
(2) the bills received in exchange must be of currently-available, printed denominations
If Camille asks for change for $3000, she will get:
$100 $50 $20 $10 $5 $2 $1
-----------------------------------------------------
? ? ? ? ? ? ?
(a) One method of finding a solution is by “bounding” the answer. We know that 30 $100 bills is $3000 (too many bills) and there is no available higher denomination, so (a) can’t be done.
(b) Camille’s $3000 could be 50 bills because 30 $100 bills is $3000 and 60 $50 bills is $3000. So, start with 30 $100 bills and begin “trading” a $100 bill for two $50 bills until you get 50 fills. The options are:
$100 $50 $20 $10 $5 $2 $1
-----------------------------------------------------
29 0 2 1 8 0 10
29 0 1 4 6 0 10
(1) "One-thousand dollar bills may seem rare and exotic to the casual observers. The latest series date is 1934A. Surprisingly though, $1,000 bills were seen in circulation up until the early 1970s. You could still order them from banks in the 1960s. It is not uncommon for older Americans to take them to the bank to try to get face value today. Of course, we wouldn’t recommend selling them for face value. Only severely impaired examples should be worth face value. All other notes are definitely collectible. Some $1,000 bills can be worth several thousand dollars each. Your standard value for a generic note in lightly circulated condition is probably $1,600. However, there are plenty of exceptions to that rule." See --http://www.antiquemoney.com/old-one-thousand-dollar-bill-value-price-guide/
(2) the bills received in exchange must be of currently-available, printed denominations
If Camille asks for change for $3000, she will get:
$100 $50 $20 $10 $5 $2 $1
-----------------------------------------------------
? ? ? ? ? ? ?
(a) One method of finding a solution is by “bounding” the answer. We know that 30 $100 bills is $3000 (too many bills) and there is no available higher denomination, so (a) can’t be done.
(b) Camille’s $3000 could be 50 bills because 30 $100 bills is $3000 and 60 $50 bills is $3000. So, start with 30 $100 bills and begin “trading” a $100 bill for two $50 bills until you get 50 fills. The options are:
$100 $50 $20 $10 $5 $2 $1
-----------------------------------------------------
29 0 2 1 8 0 10
29 0 1 4 6 0 10
29 0 0 7 4 0 10
28 2 2 2 6 0 10
28 2 2 2 6 0 10
28 2 1 5 4 0 10
28 2 0 8 2 0 10
28 2 0 0 20 0 0
28 0 8 2 2 0 10
28 0 7 5 0 0 10
28 0 6 0 16 0 0
28 0 5 3 14 0 0
28 0 4 6 12 0 0
28 0 3 9 10 0 0
28 0 2 12 8 0 0
28 0 1 15 6 0 0
28 0 0 18 4 0 0
. . . and many more
Exhaustive enumeration is a common method used by computer programs because (1) it identifies all possible solutions and (2) computers are very, very fast and very, very accurate.
Using either method, when Camille requests more bills, that further limits the number of bills of large denomination (and, thus, the number of combinations to consider), so the problem is solved faster [note: there are still only a fixed number of denominations to use].
. . . and many more
Exhaustive enumeration is a common method used by computer programs because (1) it identifies all possible solutions and (2) computers are very, very fast and very, very accurate.
Using either method, when Camille requests more bills, that further limits the number of bills of large denomination (and, thus, the number of combinations to consider), so the problem is solved faster [note: there are still only a fixed number of denominations to use].
