# Out of 100 students, two sections of 40 and 60 are formed.

Question:

Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that

(a) you both enter the same sections?

(b) you both enter the different sections?

Solution:

My friend and I are among the 100 students.

Total number of ways of selecting 2 students out of 100 students $={ }^{100} \mathrm{C}_{2}$

(a) The two of us will enter the same section if both of us are among 40 students or among 60 students.

$\therefore$ Number of ways in which both of us enter the same section $={ }^{40} \mathrm{C}_{2}+{ }^{\infty} \mathrm{C}_{2}$

$\therefore$ Probability that both of us enter the same section

$=\frac{{ }^{40} \mathrm{C}_{2}+{ }^{60} \mathrm{C}_{2}}{{ }^{100} \mathrm{C}_{2}}=\frac{\frac{\lfloor 40}{2\lfloor 38}+\frac{\lfloor 60}{\lfloor 2\lfloor 58}}{\frac{\lfloor 100}{2\lfloor 98}}=\frac{(39 \times 40)+(59 \times 60)}{99 \times 100}=\frac{17}{33}$

(b) P(we enter different sections)

= 1 − P(we enter the same section)

$=1-\frac{17}{33}=\frac{16}{33}$