Let $$[t]$$ be the greatest integer less than or equal to $$t$$. Let $$A$$ be the set of all prime factors of 2310 and $$f : A \rightarrow \mathbb{Z}$$ be the function $$f(x) = \left[\log_2\left(x^2 + \left[\frac{x^3}{5}\right]\right)\right]$$. The number of one-to-one functions from $$A$$ to the range of $$f$$ is

Solution

2310 = 2×3×5×7×11. A = {2,3,5,7,11} (5 elements). f(2)=[log₂(4+1)]=[2.32]=2. f(3)=[log₂(9+5)]=[3.81]=3. f(5)=[log₂(25+25)]=[5.64]=5. f(7)=[log₂(49+68)]=[log₂117]=[6.87]=6. f(11)=[log₂(121+266)]=[log₂387]=[8.59]=8. Range={2,3,5,6,8}: 5 elements. One-to-one from 5 to 5: 5!=120.

Option (4): 120.