Compute $\dbinom{5}{3}$.
$$\dbinom{5}{3} = \dfrac{5!}{3!2!}=\dfrac{(5\times 4)(3\times 2\times 1)}{(3\times 2\times 1)(2\times 1)}=\dfrac{5\times 4}{2\times 1}=\boxed{10}.$$
10
Level 1
Counting & Probability
Problem: Compute $\dbinom{5}{3}$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To compute $\dbinom{5}{3}$, 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 $n = 5$ and $k = 3$. We can substitute these values into the formula:

\dbinom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!}

Next, we need to calculate the factorials involved:

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
3! = 3 \times 2 \times 1 = 6
2! = 2 \times 1 = 2

Now we can substitute these factorial values back into the equation:

\dbinom{5}{3} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10

Thus, the final answer is $10$.<|END_OF_LATENT|> The final answer is $10$.