Compute $\dbinom{505}{505}$.
$\dbinom{505}{505}=\dbinom{505}{0}=\boxed{1}.$
1
Level 1
Counting & Probability
Problem: Compute $\dbinom{505}{505}$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To compute the binomial coefficient $\dbinom{n}{k}$, we use the formula $\dbinom{n}{k} = \frac{n!}{k!(n-k)!}$. In this case, we have $n = 505$ and $k = 505$. 

Substituting these values into the formula gives us:

\dbinom{505}{505} = \frac{505!}{505!(505-505)!} = \frac{505!}{505! \cdot 0!}.

We know that $0! = 1$. Therefore, we can simplify the expression:

\dbinom{505}{505} = \frac{505!}{505! \cdot 1} = \frac{505!}{505!}.

Since any non-zero number divided by itself equals 1, we conclude that:

\dbinom{505}{505} = 1.

This result aligns with the combinatorial interpretation that there is exactly one way to choose all 505 items from a set of 505 items, confirming that the final answer is 1.<|END_OF_LATENT|> The final answer is $1$.