Question

Given below are two statements:
Statement I : A bag captains 10 white and 10 red face masks which are all mixed up. The fewest number of face masks you can take from a bag without looking and be sure to get a pair of the same color is 3. 
Statement II: The minimum number of students needed in a class to guarantee that there are at least 6 students whose birthdays fall in the same month, is 61. In the light of the above statements, choose the most appropriate answer from the option given below:

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but Statement II is incorrect
• D. Statement I is incorrect but Statement II is correct

Answer 1

The Correct Option is A

To determine the correctness of the statements, we need to individually evaluate both Statement I and Statement II based on logical reasoning and simple mathematical principles.

Analysis of Statement I:

Statement I claims that you have a bag containing 10 white and 10 red face masks, mixed up together. The question is, what is the smallest number of face masks you need to take from the bag without looking to ensure you get a pair of the same color?

• If you take 1 face mask, you only have that one mask, so there can’t be a pair of the same color.
• If you take 2 face masks, they could be one white and one red (no pair of the same color), due to which there's a 50% chance they will be the same color, and a 50% chance they will be different colors.
• If you take 3 face masks, you are assured to have at least one pair of the same color. This is because you will either have a combination of two reds and one white or two whites and one red.

Therefore, the fewest number of face masks you must take to ensure a pair of the same color is indeed 3, which makes Statement I correct.

Analysis of Statement II:

Statement II states that the minimum number of students needed in a class to guarantee that at least 6 students have birthdays in the same month is 61.

There are 12 months in a year, and to find the maximum number of students that can have birthdays spread over the 12 months without having at least 6 in one month, consider assigning 5 students to each month: 5 students ≤ 12 months = 60 students.

If we add even one more student making it 61 students, at least one of the months must have 6 students (by the pigeonhole principle). Hence, Statement II is correct.

Conclusion:

Both statements are correct because they logically and correctly describe their respective scenarios.