Question: There are 7 biscuits in a tin. There are $c$ chocolate biscuits and the rest are plain. You take 2 biscuits
Find an expression in terms of $c$ for probability you pick 2 chocolate biscuits.
1. **State the problem:**
We have a total of 7 biscuits, with $c$ chocolate biscuits and the rest plain. We want the probability of picking 2 chocolate biscuits when selecting 2 biscuits.
2. **Formula for probability:**
The probability of an event is given by:
$$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$
3. **Calculate total number of ways to pick 2 biscuits from 7:**
This is a combination problem where order does not matter:
$$\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21$$
4. **Calculate number of ways to pick 2 chocolate biscuits from $c$ chocolate biscuits:**
$$\binom{c}{2} = \frac{c \times (c-1)}{2}$$
5. **Write the probability expression:**
$$\text{Probability} = \frac{\binom{c}{2}}{\binom{7}{2}} = \frac{\frac{c(c-1)}{2}}{21}$$
6. **Simplify the expression:**
$$= \frac{c(c-1)}{2} \times \frac{1}{21} = \frac{c(c-1)}{42}$$
**Final answer:**
$$\boxed{\frac{c(c-1)}{42}}$$
This is the probability of picking 2 chocolate biscuits from the tin in terms of $c$.