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Let f:{1,2,3,4,5} to {1,2,3,4} be a function randomly formed. What is the probability that f is onto and f(i)≠i , i=1,2,3,4,5Answer is given 2/9Solution with explanation is required |
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Answer» Let us first count the number of ELEMENTS in F. Total number of functions from A to B is 34=81. The number of functions which contain exactly two elements in the range is 3⋅24=48. The number of functions which contain exactly one elements in its range is 3×14=3 . THUS, the number of onto functions from A to B is 81−48+3=36 [using principle of inclusion exclusion] ∴n(F)=36 Let f∈F. We now count the number of ways in which f−1(x) consists of single ELEMENT. We can choose PREIMAGE of x in 4 ways. The remaining 3 elements can be mapped onto [y,Z] is 23−2=6 ways. ∴f−1(x) will consists of exactly one element in 4×6=24ways. Thus, the probability of the required event is 24/36=2/3. |
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