Question

To process a workpiece 10 operations are interchangeable in their order. The
economic-technological evaluation of each variant lasts 1h. How long would the overall rating take to find the optimal machine layout?

How many Morse alphabet characters can be formed when a character is out
a) exactly five elements b) should consist of at most five elements?

In the vicinity of a resort 15 hiking trails are to be marked by two colored, parallel lines.
How many colors do you need at least if the same color pairs are allowed and the arrangement of the lines does not matter?

Answer 1

1. There are a total of 10 workpiece operations that are interchangeable in there order. Meaning, we will have to search through all the permutations of the order in which the operations are performed. For 10 unique items, 10! (factorial 10) different permutations are possible. Therefore, for finding the optimal machine layout it would take 10! = 3628800 hours.

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2. In Morse Language there are only two symbols '.' and '-'. Therefore, for

a) creating a Morse character of 5 elements, we can put either '.' or '-' in each of the 5 places. So for each place we have 2 choices and altogether, therefore there are characters possible

b) in this case, the character can be either 1, 2, 3, 4, or 5 symbols long. Following the similar argument as in part (a), we can say that there are characters of 5 elements possible. Similarly, there are characters of 4 elements possible. Therefore, the number of characters possible with at most 5 elements is

. So, there are 62 such characters possible

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3. There are 15 hiking trails and each needs to be marked by two parallel lines, which may or may not be of different color. Therefore, we need to fill 30 parallel lines with colors such that each of the 15 hiking trails have a different marker.

If we have colors, then the number of patterns (if different arrangement of same color corresponds to different pattern for example, if, red-yellow and yellow-red are different patterns) we can create from them is . This is so, because we have two lines and each of these lines can be filled with colors.

However, it is given that the arrangement of lines does not matter, therefore, red-yellow and yellow-red correspond to the same marker. Now, in the markers, we known that the number of same colored markers is , 1 for each color. Therefore, total number of markers with different color is . We have to consider only half of these, because other half is only a different arrangement of the same colors.

Thus, the total number of patterns possible is

Solving for ,

since, 5^2 + 5 = 25 + 5 = 30

Therefore, there are 5 colors required.