Three coins are randomly placed into different positions on a 4×10 grid. The probability that no two coins are in the same row or column can be expressed as ab where a and b are coprime positive integers. What is the probability of a+b?
To solve this problem, we need to analyze the possible placements of the three coins on the 4×10 grid. We will start by considering the different rows and columns separately.
For the first coin, we have 40 possible positions on the grid (4 rows × 10 columns). Once the first coin is placed, there are 30 remaining positions for the second coin. However, this second coin cannot be placed in the same row or column as the first coin. Thus, we have 30 position choices for the second coin, as there are 3 rows and 10 columns that are available for placement.
For the third coin, there are now only 20 remaining positions on the grid. However, it cannot be placed in the same row or column as the first two coins. Since the first coin already occupies one row and one column, and the second coin occupies one additional row and one additional column, there are now only 2 rows and 8 columns remaining for the third coin.
To calculate the probability, we need to divide the number of favorable outcomes (i.e., valid coin placements) by the total number of possible outcomes.
The total number of possible outcomes is simply the number of ways we can place three coins on a 4×10 grid, which is given by:
Total outcomes = 40 × 30 × 20
The number of favorable outcomes is the number of ways we can place the coins such that no two coins are in the same row or column. This is given by:
Favorable outcomes = 40 × 30 × 2
Therefore, the probability can be expressed as:
Probability = Favorable outcomes / Total outcomes
Probability = (40 × 30 × 2) / (40 × 30 × 20)
Simplifying, we get:
Probability = 2 / 20
Expressed as a fraction, this can be further simplified to:
Probability = 1 / 10
So, the probability can be expressed as 1/10, giving us the values a=1 and b=10.
To find the sum a+b, we simply add the values together:
a + b = 1 + 10 = 11
Therefore, the probability of a+b is 11.