The numbers 1,2,…,17 are divided into 5 disjoint sets. One set has 5 elements, one set has 4 elements, two sets have 3 elements and the last set contains the 2 remaining elements. Two players each choose a number from 1 to 17 at random. The probability they choose numbers from the same set can be expressed as ab, where a and b are coprime positive integers. What is the value of a+b?
probability is from 0 to 1.
No way it can be expressed as the product of integers.
.76
you son of a prostitute the answer is wrong
51
To solve this problem, we need to understand the concept of disjoint sets and probability. Let's break it down step by step.
We are given that the numbers 1 to 17 are divided into 5 disjoint sets. Disjoint sets are sets that have no elements in common. In this case, we have one set with 5 elements, one set with 4 elements, two sets with 3 elements each, and one set with 2 elements.
Now, let's consider the probability that two players choose numbers from the same set. To calculate this probability, we need to find the total number of ways the players can choose numbers from the same set and divide it by the total number of possible choices they can make.
Let's calculate each step:
1. Calculating the total number of ways the players can choose numbers from the same set:
- For the set with 5 elements, each player can choose one number in 5 ways.
- For the set with 4 elements, each player can choose one number in 4 ways.
- For the two sets with 3 elements each, each player can choose one number in 3 ways.
Since the players are choosing independently, we multiply the number of choices for each set together:
5 * 4 * 3 * 3 = 180
2. Calculating the total number of possible choices the players can make:
Each player can choose any number from 1 to 17, so they have 17 options each.
Since the players are choosing independently, we multiply the number of choices for each player together:
17 * 17 = 289
3. Calculating the probability:
The probability is given by the ratio of the total number of ways the players can choose numbers from the same set to the total number of possible choices:
180/289
To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator (180) and the denominator (289). If the GCD is 1, it means the numerator and denominator share no common factors other than 1, making them coprime.
In this case, the GCD of 180 and 289 is 1, which means they are coprime.
Therefore, the probability can be expressed as ab = 180/289.
The value of a + b is 180 + 289 = 469.