There are 10 pens, including 2 black pens, 3 red pens and 5 pink pens. Take 2 pens at random. Calculate the probability that both pens are pink

Answer = To calculate the probability that both pens drawn randomly are pink, we'll use the concept of combinations. Total number of pens = 10 Number of pink pens = 5 We need to choose 2 pens out of the total. The number of ways to choose 2 pens out of 10 is given by the combination formula: \[ \text{Number of ways to choose 2 pens out of 10} = \binom{10}{2} \] \[ = \frac{10!}{2!(10-2)!} \] \[ = \frac{10 \times 9}{2 \times 1} \] \[ = 45 \] Now, we need to find the number of ways to choose 2 pink pens out of 5. This is also a combination: \[ \text{Number of ways to choose 2 pink pens out of 5} = \binom{5}{2} \] \[ = \frac{5!}{2!(5-2)!} \] \[ = \frac{5 \times 4}{2 \times 1} \] \[ = 10 \] Now, the probability of choosing 2 pink pens is the ratio of the number of successful outcomes (choosing 2 pink pens) to the total number of outcomes (choosing any 2 pens): \[ P(\text{both pens are pink}) = \frac{\text{Number of ways to choose 2 pink pens}}{\text{Number of ways to choose 2 pens out of 10}} \] \[ P(\text{both pens are pink}) = \frac{10}{45} \] \[ P(\text{both pens are pink}) = \frac{2}{9} \] So, the probability that both pens drawn randomly are pink is \( \frac{2}{9} \).