In this problem, a word is any string of 12 numbers (e.g., 002342393765 is a codeword but 01245 is not, since it consists of only five numbers). Provide a brief justification for each of your answers (no more than five lines or so), explaining which counting rules you used and what your thought process was. Feel free to give expressions in the form 2·410, etc. in your final answers; no need to use calculators to compute such powers. 1. How many codewords are there? 2. How many codewords end with 45? 3. How many codewords begin with 1 and end with 1? 4. How many codewords begin with 52 or 62? 5. How many codewords begin with 00 or end with 00? 6. How many codewords begin with 22 or 12 and end with 0 or 1? 7. How many codewords begin with a number strictly smaller than 5 and end with a number strictly larger than 5? 8. How many codewords have their first three numbers be all strictly smaller than 6? For example, 03242393765 and 01342393765 are such codewords but 02742393765 is not. 9. How many codewords have no zeros and no ones and no twos? 10. How many codewords have no ones or no fives?

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Average

In mathematics, everyone must have heard the term average in one place or other. Almost all types of calculation require taking out an average. In everyday life, the term is used for denoting a certain amount that can be said to be a typical quantity for multiple people or objects. In other words, if a group of people possesses a certain object, then the average gives us an idea regarding the quantity of the object possessed by each person. However, this is solely based on mathematical calculation. It is important to note that most people will not strictly have the average value in a group.