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The nation of Seedonia, issues license plates for all cars, trucks and motorcycles. Long ago the Ministry of Motor Vehicles (MMV) decided that every license plate would have 6 numbers, divided by a dash in the middle. Many years ago the MMV realized that they needed more license plates and launched a new system by which every license plate had 6 characters, but that two of the characters must be capital letters in the standard English alphabet. They decided not to use the letters “I” and “O” because they are too similar to numbers, “1” and “0”.
How many license plates could Seedonia’s MMV issue using six numbers?
How many license plates can the MMV issue using two letters and four numbers on every plate?
- If population and the number of vehicles grow, which of the following alternatives would be better?
- a) add one more number, for a total of 7, so each plate has 2 letters and 5 numbers
b) add one more letter, but keep the total at 6, so each plate has three letters and 3 numbers
Combinatorics is a “branch of mathematics concerned with the selection, arrangement, and combination of objects chosen from a finite set.” (Britannica Concise Encyclopedia) The number of different ways to deal a hand in a card game like poker, bridge or solitaire is a simple example (although it’s a big number). Assigning students and scheduling classes is another example. There are few standard algorithms for problems in combinatorics. Instead, each problem requires its own logical analysis—this makes combinatorics a great field for finding interesting math puzzles. The development of computer networks with so many different codes and passwords makes combinatorics one of the most important mathematical fields of our day.
For the past two months, our puzzles involved geometric combinatorics, as we tried to find all the possible nets for several different three-dimensional shapes. This month we have a more straightforward situation involving combinations of numbers and letters. Like our problems with nets and shapes, these problems also call for careful, exhaustive analysis and consideration of all cases.
Scientific notation is a system for representing large numbers and doing calculations with them rapidly. Scientific notation uses powers of ten to represent the number of zeros after the decimal point in a large number. Some examples:
1 million, 1,000,000 is represented as 1 x 106 because there are six zeros to the right of the number 1.
157, 000 is represented as 1.57 x 105, which is the same as multiplying 1.57 by 100,000.
1 or 1.57 is called the coefficient; 106 or 105 is called the base. The coefficient is always equal to or greater than 1 and always less than 10.
When you multiply powers of 10, you just add them. 105 x 103 = 108.
So to multiply two numbers expressed in scientific notation, you multiply the coefficients and add the powers of 10 in the bases.
(3.5 x 105) x (9 x 103) = 31.5 x 108 = 3.15 x 109.
When you divide numbers in scientific notation, you divide the coefficients and subtract the powers of 10 in the bases.
(3.5 x 105) / (9 x 103) = .389 x 102 = 3.89 x 101 = 38.9
You can find more detailed information about this at many different web sites.
After you’ve tried this for yourself check our solutions. |
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