1

Here is the problem in full with no other special restrictions:

"If there are 21 students in a class, show that at least 11 must be male or female."

QuIcKmAtHs
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Tech Avenue
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2 Answers2

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Assume there are at most 10 male students and at most 10 female students. Then there are at most 20 students, a contradiction.

J.G.
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  • What's this discrimination towards hermaphrodite students?! – Raskolnikov Jan 18 '18 at 10:35
  • Haha @Raskolnikov – QuIcKmAtHs Jan 18 '18 at 10:38
  • You have bought The Special Chocolate Cake with n number of slices (where n is even lets say 10 number of slices). You have only invited one special friend of yours. You would like to eat it all together but eating has some particular rules: ✓ You cannot eat randomly. The First slice has to be eaten by you. ✓ After you have started eating second person has got two choice i.e. eat it one from any two of the neighbours which you selected. ✓ Similarly after your friend took one you also have exactly two choices. at-least half of the cake must be eaten by you. – Tech Avenue Jan 18 '18 at 10:59
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Let us proof by contradiction. By assuming that there are only 10 boys and 10 girls, we attempt to not fulfill the requirement. However, there are 21 people, so there must be one more boy or girl, meaning that there would be 11 boys or 11 girls.

To answer the comment, we also proof by contradiction. Let us say there are less than 5 males. That would mean that there will be more than 21-5=16 females. Hence it is proven that there will be at least 5 males or 16 females.

QuIcKmAtHs
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  • Now prove also that class must have at least 5 male students or at least 16 female students. – Tech Avenue Jan 18 '18 at 10:37
  • @TechAvenue do you get my solution? – QuIcKmAtHs Jan 18 '18 at 10:42
  • Can i ask you another one, if you do not mind....???? – Tech Avenue Jan 18 '18 at 10:47
  • You have bought The Special Chocolate Cake with n number of slices (where n is even lets say 10 number of slices). You have only invited one special friend of yours. You would like to eat it all together but eating has some particular rules: ✓ You cannot eat randomly. The First slice has to be eaten by you. ✓ After you have started eating second person has got two choice i.e. eat it one from any two of the neighbours which you selected. ✓ Similarly after your friend took one you also have exactly two choices. at-least half of the cake must be eaten by you. – Tech Avenue Jan 18 '18 at 10:58
  • Please post this as another post, not a comment – QuIcKmAtHs Jan 18 '18 at 10:58
  • i am not allowed to post more than 2 questions – Tech Avenue Jan 18 '18 at 10:59
  • Hmm... okay... but what is your above comment-question asking? – QuIcKmAtHs Jan 18 '18 at 11:01
  • Yeah, its a question again about pigeonhole principle..... – Tech Avenue Jan 18 '18 at 11:03
  • I am sorry. What is the question? I don't see an explicit question statement – QuIcKmAtHs Jan 18 '18 at 11:04
  • You have bought The Special Chocolate Cake with n number of slices (where n is even lets say 10 number of slices). You have only invited one special friend of yours. You would like to eat it all together but eating has some particular rules: ✓ You cannot eat randomly. The First slice has to be eaten by you. ✓ After you have started eating second person has got two choice i.e. eat it one from any two of the neighbours which you selected. ✓ Similarly after your friend took one you also have exactly two choices. at-least half of the cake must be eaten by you. – Tech Avenue Jan 18 '18 at 11:05
  • You see, what you have just sent here has no question statement. What are you asking for? The number of ways? – QuIcKmAtHs Jan 18 '18 at 11:07
  • yeah....the number of ways in which i can eat the half of the cake – Tech Avenue Jan 18 '18 at 11:11
  • There are only 4 possible pieces of cake which you can eat: 1, 2, 9,10. – QuIcKmAtHs Jan 18 '18 at 11:23