Question

Out of 100 students, two sections of 40 and 60 students are formed, if you and your friends are among the 100 students. What is the probability that

(i) you both enter the same section?

(ii) you both enter the different section?

Answer 1

Out of 100 students, two sections of 40 and 60 students can be formed in

${}^{100}{C}_{40}{\times }^{60}{C}_{60}=\frac{100!}{40!60!}$ ways

(i) If you and your friend can be in the same section, the number of ways

${}^{98}{C}_{38}{\times }^{60}{C}_{60}{+}^{98}{C}_{58}{\times }^{40}{C}_{40}=(\frac{98!}{60!38!}+\frac{98!}{58!40!})$ ways

$\therefore$ Required probability = $\frac{\frac{98!}{60!38!}+\frac{98!}{58!40!}}{\frac{100!}{60!40!}}=\frac{40\times 39}{100\times 99}+\frac{60\times 59}{100\times 99}$

= $\frac{26}{165}+\frac{59}{165}=\frac{85}{165}=\frac{17}{33}$

(ii) probability of entering in the different sections = 1 - Probability of entering the same section

$\therefore$ Required probability = $1-\frac{17}{33}=\frac{16}{33}$