Probability

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=‌

50×49×48

6

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=‌

25×24

2

Similarly, the number of ways to choose two odd numbers is also:
‌25C2=‌

25×24

2

Thus, the total number of favorable cases (either two evens or two odds) is:
2×‌25C2=2×‌

25×24

2

Therefore, the probability that the selected numbers are in arithmetic progression is:
‌ Probability ‌=‌

2× 25×24 24

50×49×48 6

=‌

3

98