TS EAMCET 10 May 2024 Shift 2 Solved Paper

When selecting three numbers from the set $\{1,2,3,\dots,50\}$, 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:${}^{50}C_3 = \frac{50 \times 49 \times 48}{6}$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 $\{1,2,3,\dots,50\}$ consists of 25 odd numbers and 25 even numbers.The number of ways to choose two even numbers is:${}^{25}C_2 = \frac{25 \times 24}{2}$Similarly, the number of ways to choose two odd numbers is also:${}^{25}C_2 = \frac{25 \times 24}{2}$Thus, the total number of favorable cases (either two evens or two odds) is:$2 \times {}^{25}C_2 = 2 \times \frac{25 \times 24}{2}$Therefore, the probability that the selected numbers are in arithmetic progression is:$\text{Probability} = \frac{2 \times \frac{25 \times 24}{24}}{\frac{50 \times 49 \times 48}{6}} = \frac{3}{98}$