Question

From out of 100 enrolled students, two sections of strength 40 and 60 are formed. If you and your friend are among those 100 students, then the probability that both of you are placed in the same section is

• A. \( \frac{{}^{98}C_{40} + {}^{98}C_{60}}{{}^{100}C_{40}} \)
• B. \( \frac{{}^{40}C_{2} + {}^{60}C_{2}}{{}^{100}C_{2}} \)
• C. \( \frac{{}^{98}C_{60} + {}^{98}C_{38}}{{}^{100}C_{60}} \)
• D. \( \frac{{}^{98}C_{2} + {}^{0}C_{2}}{{}^{100}C_{2}} \)

Hint:

Consider the positions of you and your friend within the 100 students. The total number of ways you and your friend can be placed is equivalent to choosing 2 students out of 100, which is \( {}^{100}C_{2} \). For the favorable outcomes, consider the case where both of you are in the section of 40 and the case where both of you are in the section of 60. The number of ways for each case is \( {}^{40}C_{2} \) and \( {}^{60}C_{2} \) respectively. The probability is the sum of these favorable outcomes divided by the total number of outcomes.

Answer 1

The Correct Option is B

Total number of students = 100.
Two sections are formed: Section 1 with 40 students and Section 2 with 60 students.
You and your friend are among the 100 students.
We want to find the probability that both of you are in the same section.
Total number of ways to choose 2 students out of 100 (you and your friend) is \( {}^{100}C_{2} \).
This is the total number of possible pairs you and your friend can form within the group of 100.
Case 1: Both of you are in Section 1 (strength 40).
Number of ways to choose the remaining \( 40 - 2 = 38 \) students from the other \( 100 - 2 = 98 \) students to be in Section 1 with you and your friend.
However, since we are only concerned about the probability of you and your friend being together, we can directly consider the number of ways to place you and your friend in Section 1.
If both of you are in Section 1, then the remaining 38 students in Section 1 can be any of the other 98 students.
The number of ways to choose 2 spots out of the 40 spots in Section 1 for you and your friend is \( {}^{40}C_{2} \).
Case 2: Both of you are in Section 2 (strength 60).
Similarly, if both of you are in Section 2, then the remaining 58 students in Section 2 can be any of the other 98 students.
The number of ways to choose 2 spots out of the 60 spots in Section 2 for you and your friend is \( {}^{60}C_{2} \).
The number of ways that both of you are in the same section is the sum of the number of ways in Case 1 and Case 2: \( {}^{40}C_{2} + {}^{60}C_{2} \).
The probability that both of you are placed in the same section is the ratio of the number of favorable outcomes to the total number of possible outcomes: \[ P(\text{both in same section}) = \frac{ \text{Number of ways both are in Section 1} + \text{Number of ways both are in Section 2} }{ \text{Total number of ways to place you and your friend} } \] $$ P(\text{both in same section}) = \frac{{}^{40}C_{2} + {}^{60}C_{2}}{{}^{100}C_{2}} $$