Question
(a) How many students must be in a room to be sure that at least two are the same sex? (b) How many students must be in a room to be sure that at least nine are boys or at least nine are girls? (a) There must be at least \( \square \) students in a room to ensure there are at least 2 of the same sex. (b) There must be at least \( \square \) students in a room to ensure there are at least nine boys or nine girls.
Ask by Ward Huang. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**(a)**
To ensure at least two students are the same sex, there must be at least **3** students in the room.
**(b)**
To ensure there are at least nine boys or nine girls, there must be at least **17** students in the room.
Solution
**(a)**
We consider the two sexes as two groups (boys and girls). In the worst-case scenario, if there are only two students, they could be of different sexes—one boy and one girl. To ensure that at least two students belong to the same sex, we must add one more student. Thus, by the pigeonhole principle, having
\[
3
\]
students guarantees that at least two are of the same sex.
**(b)**
We want to guarantee that there are at least nine boys or at least nine girls. Consider the worst-case scenario where the numbers are as balanced as possible but still avoid having nine of either: the maximum number of each without reaching nine is \(8\). With \(8\) boys and \(8\) girls, there are
\[
8 + 8 = 16
\]
students, and neither group has reached nine. Therefore, adding one more student (making a total of \(17\) students) forces one group to have at least nine students. Thus, we need
\[
17
\]
students to ensure there are at least nine boys or at least nine girls.
**Answers:**
(a) There must be at least \(\boxed{3}\) students in a room to ensure there are at least 2 of the same sex.
(b) There must be at least \(\boxed{17}\) students in a room to ensure there are at least nine boys or nine girls.
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Simplify this solution The Deep Dive
(a) To guarantee that at least two students are of the same sex, you only need three students in the room. This is based on the pigeonhole principle: with two categories (boys and girls), adding a third person inevitably results in at least one category having at least two individuals. (b) For ensuring that there are at least nine boys or nine girls, you need a minimum of 17 students. If you have 8 boys and 8 girls, that's only 16 in total. Introducing one more student guarantees that at least one group will reach nine. Think of it as a friendly race to nine!
