From a group of 50 students, two sections comprising of 20 and 30 students are formed. If Ram and Rahim are…
From a group of 50 students, two sections comprising of 20 and 30 students are formed. If Ram and Rahim are two particular students among the 50 students, then the probability that they both belong to the same section is
\(\frac{25}{49}\)
\(\frac{12}{23}\)
\(\frac{13}{23}\)
\(\frac{24}{49}\)
Solution
Total number of students \(=50\)
Number of ways of making groups if both the students are in first group \(={ }^{48} C_{18} \times{ }^{30} C_{30}={ }^{48} C_{18}\) and number of ways of making groups if both the students are in second group
\(={ }^{48} C_{28} \times{ }^{20} C_{20}={ }^{48} C_{28}\)
and total number of ways \(={ }^{50} C_{20} \times{ }^{30} C_{30}={ }^{50} C_{20}\)
\(\begin{gathered}
\therefore \text { Required probability }=\frac{{ }^{48} C_{18}+{ }^{48} C_{28}}{{ }^{50} C_{20}} \\
=\frac{\frac{48 !}{18 ! 30 !}+\frac{48 !}{28 ! 20 !}}{\frac{50 !}{20 ! 30 !}} \\
=\frac{\frac{1}{30 \times 29}+\frac{1}{20 \times 19}}{\frac{50 \times 49}{20 \times 19 \times 30 \times 29}}=\frac{(20 \times 19)+(30 \times 29)}{(50 \times 49)} \\
=\frac{38+87}{5 \times 49}=\frac{125}{5 \times 49}=\frac{25}{49}
\end{gathered}\)
Hence, option (a) is correct.