Compute $\dbinom{1293}{1}$.
$\dbinom{1293}{1} = \dfrac{1293!}{1!1292!}=\dfrac{1293}{1}=\boxed{1293}.$
1293
Level 1
Counting & Probability
Problem: Compute $\dbinom{1293}{1}$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To compute $\dbinom{1293}{1}$, we need to understand the definition of the binomial coefficient. The binomial coefficient $\dbinom{n}{k}$ represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to the order of selection. The formula for the binomial coefficient is given by:

\dbinom{n}{k} = \frac{n!}{k!(n-k)!}

In this case, we have $n = 1293$ and $k = 1$. Substituting these values into the formula gives:

\dbinom{1293}{1} = \frac{1293!}{1!(1293-1)!}

This simplifies to:

\dbinom{1293}{1} = \frac{1293!}{1! \cdot 1292!}

Next, we can simplify further. The factorial $1293!$ can be expressed as $1293 \times 1292!$. Thus, we have:

\dbinom{1293}{1} = \frac{1293 \times 1292!}{1 \times 1292!}

The $1292!$ in the numerator and denominator cancels out, leading to:

\dbinom{1293}{1} = \frac{1293}{1} = 1293

Therefore, the final answer is $1293$.<|END_OF_LATENT|> The final answer is $1293$.