Sets, Relations & Functions
Counting One-One Functions
nta_pyq_2024_apr
Grade 11
Let $[t]$ be the greatest integer $\leq t$. Let $A$ be the set of all prime factors of 2310 and $f:A\to\mathbb{Z}$ be the function $f(x)=\left[\log_2\left(x^2+\left[\dfrac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is:
Step-by-Step Solution
Key Concept: $2310=2\times3\times5\times7\times11$. $A=\{2,3,5,7,11\}$. Compute $f$ on each: $f(2)=2,f(3)=3,f(5)=5,f(7)=6,f(11)=8$. Range $B=\{2,3,5,6,8\}$, $|B|=5$.
Range has 5 elements. One-one functions $=5!=120$.
Correct Answer: 4