A bag contains $n$ metal coins, $n \geq 3$, that are made from either silver or bronze - HSC - SSCE Mathematics Extension 1 - Question 9 - 2024 - Paper 1

To find the probability that the two coins drawn are of the same metal, we will sum the probabilities of drawing two silver coins and two bronze coins.

1. Probability of drawing 2 silver coins:

   • The number of ways to choose 2 silver coins from $k$ is given by $\binom{k}{2} = \frac{k(k-1)}{2}$.

2. Probability of drawing 2 bronze coins:

   • The number of ways to choose 2 bronze coins from $(n - k)$ is given by $\binom{n - k}{2} = \frac{(n-k)(n-k-1)}{2}$.

The total number of ways to choose any 2 coins from $n$ coins is given by $\binom{n}{2} = \frac{n(n-1)}{2}$.

The total favorable outcomes for drawing 2 coins of the same metal is:

$\binom{k}{2} + \binom{n-k}{2} = \frac{k(k-1)}{2} + \frac{(n-k)(n-k-1)}{2}$

Thus, the probability that the two coins drawn are of the same metal is:

$P = \frac{\frac{k(k-1)}{2} + \frac{(n-k)(n-k-1)}{2}}{\frac{n(n-1)}{2}} = \frac{k(k-1) + (n-k)(n-k-1)}{n(n-1)}$

This matches option A.