<s>[INST] The positive integer $n$ has the prime factorization $p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$. We can factor $n$ as the product of $n_1,n_2,\ldots,n_m$ as follows:
$n=n_1n_2\cdots n_m=p_1^{e_{11}}p_2^{e_{21}}\cdots p_k^{e_{k1}} \cdot p_1^{e_{12}}p_2^{e_{22}}\cdots p_k^{e_{k2}} \cdots p_1^{e_{1m}}p_2^{e_{2m}}\cdots p_k^{e_{km}}$
We can see that $e_{ij}$ is a non-negative integer and $e_{i1}+e_{i2}+\cdots+e_{im}=e_i$ for all $i$.
Suppose we choose $m$ factors of $n$ randomly and reassemble them into a new integer $n'$. What is the probability that the number $n'$ (the product of the chosen $m$ factors)
has the same prime factorization as $n$?
 My question is how to read the lines and how to approach this problem.[/INST]