The squares of a 3×3 grid of unit squares are coloured randomly and independently so that each square gets one of 5 colours. Three points are then chosen uniformly at random from inside the grid. The probability that these points all have the same colours can be expressed as a/b, where a and b are coprime positive integers. What is the value of a+b?
Have you got the answer??
I'm so confused..
I don't know how to solve
is it 458?
no not 458
so what's the answer?
give me the solution..
1528
To solve this problem, we need to calculate the probability that three randomly chosen points from the grid have the same color.
First, let's consider the total number of possible color combinations for the three chosen points. Since each square can have one of 5 colors, there are a total of 5 * 5 * 5 = 5^3 = 125 possible color combinations.
Next, we need to count the number of favorable outcomes, i.e., the number of ways we can choose three points with the same color. To do this, we consider each color separately.
For a single color, there are 9 possible ways to choose three points with that color:
- Choose all three points from one row (3 ways)
- Choose all three points from one column (3 ways)
- Choose all three points from one diagonal (3 ways)
Since there are 5 colors, the total number of favorable outcomes is 9 * 5 = 45.
Now we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
probability = favorable outcomes / total outcomes = 45 / 125 = 9 / 25.
The value of a+b is therefore 9 + 25 = 34.