Question: There are 10 pens in a box.
There are $x$ red pens in the box.
All the other pens are blue.
Jack takes at random two pens from the box.
Find an expression, in terms of $x$, for the probability that Jack takes one pen of each colour.
Give your answer in its simplest form.
1. **State the problem:**
We have a box with 10 pens, $x$ of which are red and the rest are blue. Jack randomly picks 2 pens. We want the probability that he picks exactly one red and one blue pen.
2. **Identify total pens and colors:**
- Total pens: 10
- Red pens: $x$
- Blue pens: $10 - x$
3. **Total ways to pick 2 pens:**
The total number of ways to choose any 2 pens from 10 is given by the combination formula:
$$\binom{10}{2} = \frac{10 \times 9}{2} = 45$$
4. **Ways to pick one red and one blue pen:**
We must pick 1 red pen from $x$ red pens and 1 blue pen from $10 - x$ blue pens.
Number of ways:
$$\binom{x}{1} \times \binom{10 - x}{1} = x \times (10 - x)$$
5. **Probability formula:**
$$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{x(10 - x)}{45}$$
6. **Simplify the expression:**
The expression is already in simplest form:
$$\boxed{\frac{x(10 - x)}{45}}$$
This is the probability that Jack picks one pen of each color when he randomly selects two pens from the box.