reducing fractions with combinations and powers of large integers

Question

For example, I'm trying to reduce $\frac{49 \choose6}{49^6}$. Is there anyway to reduce this fraction further? Thanks

Answer 1

You should use the formula to express the $49 \choose 6$ in terms of factorials. In this specific case, the denominator is $7^{12}$ so you only care about powers of $7$ in the numerator. Using Legendre's formula you can find that $49 \choose 6$ has a factor $7^2$ so you can divide that out. But is $$\frac {\frac 1{49}{49 \choose 6}}{49^5}$$ more enlightening than your expression? Simplification is in the eye of the beholder and yours prompts me to think of a lottery that draws six numbers of $49$.