In how many ways you can chose a committee of three students from a class of ten students?

The renewed interest in this old post, reminded me of a more recent event:

Question: In how many ways you can chose a committee of three students from a class of ten students?

My expected answer: ${10} \choose {3}$ which is 120.

Alternative answer 1:(Lior)  There are various ways: you can use majority vote, you can use dictatorship (e.g. the teacher chooses); approval voting, Borda rule…

Alternative answer 2: There are precisely four ways: with repetitions where order does not matter; with repetitions where order matters; without repetitions where order matters; without repetitions where order does not matter,

Alternative answer 3: The number is truly huge. First we need to understand in how many ways we can choose the class of ten students to start with. Should we consider the entire world population? or just the set of all students in the world, or something more delicate? Once we choose the class of ten students we are left with the problem of chosing three among them.

The Mystery Beeping Riddle

We came back from the airport with our daughter who has just landed after a four-month trip to India. The car was making a strange beep every so often.

Maybe it is an indicator signal that should have turned off automatically? No, this possibility was quickly eliminated.

I looked in the car manual. The only slightly similar symptom described there was a beeping indicating that the air bags are out of order and the air bag light warning signal is also out of order. Was this the reason? In this case there would be a 5-second beep every minute. But our beeps were once every 5 minutes and each beep was for one second. Was there some mistake in the translation of the manual to Hebrew?

I called the garage. Yes, they told me, if I bring the car they can check out what is wrong and fix it. No, they have not encountered this problem before. No, it is not dangerous to drive the car back to Jerusalem. And no, they were not familiar with translation problems in the manual.

Another breakthrough idea! Maybe the beeping came from a mobile phone in the car. Some mobile phones tend to beep when the battery is low or when there is an unread message. We turned off the two mobile phones in the car. This looked promising, Continue reading

Greg’s Dinosaurs Riddle

The two-riddles post was a success, and while corresponding with Greg Kuperberg he had a riddle for me about dinosaurs, and he agreed I will share it with you.

Right before the Chixculub asteroid hit the earth, there were a variety of mammals and a wide variety of dinosaurs.  Both the mammals and the dinosaurs lived in many places, came in many sizes, etc.  Why did some of the mammals survive, but none of the dinosaurs did?

A nuch simpler question: Can you answer (without clicking or googeling or so):

How many years did the dinosaurs live? (you know,.. not a single one but them as a whole…) 1 million years, 5 millions, 20 millions, 50 millions, 100 millions?

And a deep philosophical question:

The dinosaurs perished. Should we regard them as losers???

Two Math Riddles

Usually I am not particularly good at telling math riddles or solving them, and I was not planning on presenting math riddles here. But these days, when mathematical blogs break new ground and enter uncharted territories, let me make an exception and tell you two riddles.

1) (Told to me by Sergiu Hart; taken from a MSRI news-teller to the best of my memory.)

You have a 1 meter long (one ant thin) wooden stick and you put on it n ants in arbitrary locations. Each ant is facing one end of the stick. The ants move 1 meter per minute and when two ants meet they both start moving in the opposite directions. If an ant reaches an end of the stick it falls from it.

After how much time you can be sure that no ants will remain on the stick?

2) (Told to me (with the solution) most recently by Sasha Barvinok, who heard the problem on “Car talk“.)

An airplane has 100 seats, and 100 passengers were assigned seats. The first passenger, ‘Joe,’ enters the plane and rather than sitting in his assigned place, he sits in a random place. The next passengers come one by one and every passenger sits in his assigned seat if it is empty, and in a random empty seat if his seat is already taken. What is the probability that the last passenger ‘Jim’ will sit in his assigned seat?

If you want to solve these riddles on your own do not read the rest of this entry. Continue reading