Mathematics - Permutation & Combination Question with Solution | TestHub

Let $P$ be a prime number and $n$ be a natural number. The exponent of $P$ in $n!$ is denoted by $E_P(n!)$ and is given by:

$$E_P(n!)=\lfloor \frac{n}{P}\rfloor +\lfloor \frac{n}{P^2}\rfloor +\lfloor \frac{n}{P^3}\rfloor +\cdots +\lfloor \frac{n}{P^k}\rfloor$$

such that $P^k\le n<P^{k+1}$. Here, $\lfloor y\rfloor$ denotes the integral part of $y$.

If we isolate the power of each prime number contained in any number $N$, then we can write $N$ as $N=2^{{\alpha }_1}\cdot 3^{{\alpha }_2}\cdot 5^{{\alpha }_3}\dots$, where ${\alpha }_i$ is a natural number for $i=1,2,\dots$.

On the basis of the above information, answer the following questions.