✔ 最佳答案
We can solve this question by consider its simplified form.
Consider the form with 1 dot position.
Number of symbols have 0 raised dots = 1
Number of symbols have 1 raised dots = 1
Total number of possible symbols = 1+1 = 2
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Consider the form with 2 dot positions.
Number of symbols have 0 raised dots = 1
Number of symbols have 1 raised dots = 2
Number of symbols have 2 raised dots = 1
Total number of possible symbols = 1+2+1 = 4 = 2^2
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Consider the form with 3 dot positions.
Number of symbols have 0 raised dots = 1
Number of symbols have 1 raised dots = 3
Number of symbols have 2 raised dots = 3
Number of symbols have 3 raised dots = 1
Total number of possible symbols = 1+3+3+1 = 8 = 2^3
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Notice that the pattern of "Pascal's Triangle" is seen.
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Consider the form with 4 dot positions.
Number of symbols have 0 raised dots = 1
Number of symbols have 1 raised dots = 4
Number of symbols have 2 raised dots = 6
Number of symbols have 3 raised dots = 4
Number of symbols have 4 raised dots = 1
Total number of possible symbols = 2^4 = 16
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Consider the form with 5 dot positions.
Number of symbols have 0 raised dots = 1
Number of symbols have 1 raised dots = 5
Number of symbols have 2 raised dots = 10
Number of symbols have 3 raised dots = 10
Number of symbols have 4 raised dots = 5
Number of symbols have 5 raised dots = 1
Total number of possible symbols = 2^5 = 32
2009-01-21 19:21:33 補充：
Now, consider the form with 6 dot positions, the true form of Braille system.
Number of symbols have 0 raised dots = 1
Number of symbols have 1 raised dots = 6
Number of symbols have 2 raised dots = 15
Number of symbols have 3 raised dots = 20
2009-01-21 19:21:43 補充：
Number of symbols have 4 raised dots = 15
Number of symbols have 5 raised dots = 6
Number of symbols have 6 raised dots = 1
Total number of possible symbols = 2^6 = 64
2009-01-21 19:21:51 補充：
a) The numbers of symbols have 3 raised dots = 20
b) The numbers of symbols have an even number of raised dots = 15+15+1 = 31 (If 0 is excluded) or 1+15+15+1 = 32 (If 0 is included)
