Let ๐ and ๐ denote the sets containing 2 and 20 distinct objectsโฆ
2015
Let ๐ and ๐ denote the sets containing 2 and 20 distinct objects respectively and ๐น denote the set of all possible functions defined from ๐ to ๐. Let ๐ be randomly chosen from ๐น. The probability of ๐ being one-to-one is ________.
Answer: 0.95 โ Key idea: count all possible functions and count the one-to-one (injective) functions, then form their ratio. Total number of functions from X to Y: Each ofโฆ
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Correct answer: 0.95
Key idea: count all possible functions and count the one-to-one (injective) functions, then form their ratio.
Total number of functions from X to Y:
Each of the 2 elements of X can be mapped to any of the 20 elements of Y, so total = 20^2 = 400.
Number of one-to-one (injective) functions:
Choose distinct images for the 2 elements of X: 20 choices for the first, 19 for the second, so injective count = 20 ร 19 = 380.
Probability that a randomly chosen function is one-to-one:
Probability = 380 / 400 = 19 / 20 = 0.95.
Answer: 19/20 = 0.95.
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