easy math problem by explanation

label Algebra
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schedule 1 Day
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Computer Code.  The ASCII code used by most computers uses the last seven positions of an eight-bit byte to represent all the characters on a standard keyboard.  how many different orderings of 0's and 1's (or how many different characters) can be made by using the last seven positions of an eight-bit byte?

May 23rd, 2015


Overview: I'm going to to briefly explain the Fundamental Counting Principle and then we'll apply it to this problem.

The Fundamental Counting Principle (hereafter, FCP) tells us how many ways a series of successive events can occur. To calculate the number of ways that we can arrange a sequence of symbols we multiply the number of possible symbols for each place. 

For example if we needed a single letter code (not case sensitive) we could do it 26 ways because there are 26 letters in the alphabet.   So, our code might be a  or t.  

If we needed a two symbol letter code, there are 26 ways to choose the first value and 26 ways to choose the second value for a total of 26x26= 676 possible codes.  So, our code might be  gu or ry or hh.

For the computer code, we really only have two possibilities for each place, one (1) or zero (0). So the first symbol of our seven digit string could be either 1 _ _ _ _ _ _     or   0 _ _ _ _ _ _.   The same is true for our second place, either 1 or 0,  _ 1 _ _ _ _ _    or  _ 0 _ _ _ _ _. 

So far we have these possibilities  11_ _ _ _ _     10_ _ _ _ _     01_ _ _ _ _    00_ _ _ _ _

These four possibilities are accounted for by taking 2x2 = 4, because each of the two places place has two possibilities.

For the entire seven digit string we would have   2x2x2x2x2x2x2 = 2^7 = 128 total possible characters.

I hope that helps.  Please do ask if you have any questions or need a clarification for any step.


May 24th, 2015

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