Question

a box contains 3 red balls, 5 yellow balls and 2 green balls. two balls are chosen. what is the probability of 2 balls of the same colour?

Explanation:

Step1: Calculate total number of balls

Total balls = 3 (red) + 5 (yellow) + 2 (green) = 10.
Number of ways to choose 2 balls from 10: $C(10,2)=\frac{10!}{2!(10 - 2)!}=\frac{10\times9}{2\times1}=45$.

Step2: Calculate ways to choose 2 red balls

Number of ways to choose 2 red balls: $C(3,2)=\frac{3!}{2!(3 - 2)!}=\frac{3\times2!}{2!\times1!}=3$.

Step3: Calculate ways to choose 2 yellow balls

Number of ways to choose 2 yellow balls: $C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4\times3!}{2!\times3!}=\frac{5\times4}{2\times1}=10$.

Step4: Calculate ways to choose 2 green balls

Number of ways to choose 2 green balls: $C(2,2)=\frac{2!}{2!(2 - 2)!}=1$.

Step5: Calculate total favorable outcomes

Total favorable (same color) = 3 + 10 + 1 = 14.

Step6: Calculate probability

Probability = $\frac{\text{Favorable outcomes}}{\text{Total outcomes}}=\frac{14}{45}$.

Answer:

$\frac{14}{45}$ (corresponding to the option with $\boldsymbol{\frac{14}{45}}$)