Question 9 | BetaMath

Question 9

A bag contains n metal coins, $n \geq 3$ , that are made from either silver or bronze. There are $k$ silver coins in the bag and the rest are bronze.

Two coins are to be drawn at random from the bag, with the first coin drawn not being replaced before the second coin is drawn.

Which of the following expressions will give the probability that the two coins drawn are made of the same metal?

A. $\displaystyle \frac{k(k-1) + (n-k)(n-k-1)}{n(n-1)}$

B. $\displaystyle \binom{n}{2}\binom{n}{k}\left(1-\frac{k}{n}\right)^{n-2}$

C. $\displaystyle \frac{\binom{k}{2} + \binom{n - k}{2}}{n(n-1)}$

D. $\displaystyle \frac{k^2 + (n-k)^2}{n^2}$

Solution

The number of ways we can choose two coins is to pick any of the $n$ coins, followed by any of the $n-1$ remaining coins. This means there are $n(n-1)$ ways we can pick a pair of coins.

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