When selecting three numbers from the set
{1,2,3,...,50}, we want to find the probability that they form an arithmetic progression (AP).
First, determine the total number of ways to choose three numbers from the set:
‌50C3=‌For the numbers
a,b, and
c to be in AP,
2b=a+c, which implies that
a+c must be even. Therefore,
a and
c must both be even or both be odd.
The set
{1,2,3,...,50} consists of 25 odd numbers and 25 even numbers.
The number of ways to choose two even numbers is:
‌25C2=‌Similarly, the number of ways to choose two odd numbers is also:
‌25C2=‌Thus, the total number of favorable cases (either two evens or two odds) is:
2×‌25C2=2×‌Therefore, the probability that the selected numbers are in arithmetic progression is:
‌ Probability ‌=‌=‌