Very urgent
R_scott
answered
there are 14 consecutive pairs from 15 consecutive numbers
probability that a pair is NOT consecutive ... (105 - 14) / 105 = 91 / 105
three numbers is three distinct pairs
probability of no consecutive pairs in three numbers ... (91 / 105)^3
Explain Bot
answered
The total number of possibilities is given by the combination formula:
nCr = n! / (r!(n-r)!)
In this case, n = 15 (the total number of choices) and r = 3 (the number of selections).
Total number of possibilities = 15C3 = 15! / (3!(15-3)!) = 455
Now, let's calculate the number of ways to choose three numbers such that no two numbers are consecutive.
We can approach this problem using the principle of inclusion-exclusion:
Step 1: Calculate the total number of possibilities when there are no restrictions on consecutive numbers.
There are 15 choices for the first number, 14 choices for the second number (excluding the one adjacent to the first), and 13 choices for the third number (excluding the two adjacent to the second).
Total number of possibilities with no consecutive numbers = 15 * 14 * 13 = 2730
Step 2: Calculate the number of possibilities where two numbers are consecutive.
To do this, we have to consider all possible pairings of consecutive numbers. There are 14 pairs of consecutive numbers in the range 1 to 15. For each pairing, there are 13 choices for the third number (excluding any adjacent numbers to the pair).
Total number of possibilities with two consecutive numbers = 14 * 13 = 182
Step 3: Calculate the number of possibilities where three numbers are consecutive.
There are 13 possible sets of three consecutive numbers in the range 1 to 15.
Total number of possibilities with three consecutive numbers = 13
Now, let's calculate the number of ways to choose three numbers such that no two numbers are consecutive:
Number of ways to choose three numbers with no consecutive numbers = Total number of possibilities with no consecutive numbers - Total number of possibilities with two consecutive numbers - Total number of possibilities with three consecutive numbers
= 2730 - 182 - 13
= 2535
Finally, we can calculate the probability by dividing the number of ways to choose three numbers with no consecutive numbers by the total number of possibilities:
Probability = Number of ways to choose three numbers with no consecutive numbers / Total number of possibilities
= 2535 / 455
= 253/91
Therefore, the probability that no two numbers are consecutive is 253/91.
