Report on 25 November fortnightly prize question

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Sharif Quansah, Mugisha Uwiragiye, and Hamse Adam all got this at least mostly right.

A maths teacher has 30 students in her class, and 30 chairs in the classroom. How many possible seating plans does she have?

Answer: 30P30, or 30!, which is about 2.65×1032. In other words, 30 choices for which student to put into seat 1 × 29 possibilities for which student to put into seat 2 × 28 possibilities to put into seat 3 × …. × 1 possibility for which student to put into seat 30.

If she tries a new seating plan each day, how many days will it take for her to try all the possible plans?

Answer: 2.65×1032. This is much longer than the life of the universe. Even if the teacher tries a new seating plan each millisecond, the life of the universe so far is only about 4×1020 milliseconds, so the teacher would need about a thousand billion times longer than the age of the universe to try them all out.

The SLT have mercy and allow the teacher to select six students to move to another class. How many possible sets of six can she choose? Now she has 24 students and 30 chairs, what is the new number of possible seating plans?

Answer:30C6, or 593,775 sets of six; 30P24, or 30!/6!, or about 3.68×1029 seating plans for the remaining 24. If you imagine 24 students and six “empty seat” signs on the 30 seats, then there are 30! ways of arranging them, but since one “empty seat” sign is the same as another, we have to divide by 6! to get the number of different ways. This is still so many possibilities that the teacher would need a billion times longer than the age of the universe to try them all out at the rate of one every millisecond.