Animated Solution for Mathematics - Functions: Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f:A\to \mathbb{Z}$ be the function $f(x)=[{\log }_2(x^2+[\frac{x^3}{5}])]$. The number of one-to-one functions from $A$ to the range of $f$ is

Welcome, future engineer! Today, we are going to embark on a journey through a problem that beautifully weaves together number theory, function analysis, and combinatorics. It is a classic JEE-style problem that tests not just your calculation skills, but your ability to see the hidden structure in a mathematical expression.

Let us begin with the number $2310$. To understand the set $A$, we must find its prime factors.

We start by dividing $2310$ by $2$, which gives us $1155$. Then, $1155$ divided by $3$ is $385$. Dividing $385$ by $5$ gives us $77$, and $77$ is simply $7\times 11$.

Thus, the prime factorization is:

Our set $A$ is therefore $\{2,3,5,7,11\}$, and it contains exactly $5$ elements. This count is our first milestone.

Now, we encounter the function $f(x)=[{\log }_2(x^2+[\frac{x^3}{5}])]$. This looks intimidating, but let us break it down as a machine that processes an input $x$ through a nested structure.

For $x=2$: Since $2^2<5<2^3$, ${\log }_2(5)$ is $2.something$, so $f(2)=2$.

For $x=3$: Since $2^3<14<2^4$, ${\log }_2(14)$ is $3.something$, so $f(3)=3$.

For $x=5$: Since $2^5<50<2^6$, ${\log }_2(50)$ is $5.something$, so $f(5)=5$.

For $x=7$: Since $2^6<117<2^7$, ${\log }_2(117)$ is $6.something$, so $f(7)=6$.

For $x=11$: Since $2^8<387<2^9$, ${\log }_2(387)$ is $8.something$, so $f(11)=8$.

We have successfully mapped every element of $A$ to the range $R=\{2,3,5,6,8\}$. Notice that all five outputs are distinct.

We have a domain of $5$ elements and a codomain (the range) of $5$ elements. The question asks for the number of one-to-one functions from $A$ to $R$.

Since both sets have $5$ elements, a one-to-one function is simply a permutation of these $5$ elements. The number of such functions is:

We have reached the summit! The final answer is 120. Keep this logical flow in your toolkit, and you will conquer any problem that comes your way.