Homework Problem (MC Chap2, Prob 14) In how many ways can you place a white and a black king on a chessboard so that they can't capture each other?
Answer: 3612 (4.60+24.58+36.55)
Instructor Notes: First insight is that the number of squares attacked depends on the position of the king. Second, kids should correctly "add" the different scenarios.
Note: The symbol "." is used as a multiplication sign below
Principal 1: If the thing we are counting is an outcome of a multistage process, then the number of outcomes is the product of the number of choices for each stage
Principal 2: If the thing we are counting can happen in different exclusive ways, then the number of outcomes is the sum of the number of outcomes through each way
(BerkeleyMC Chap 2 - Prob 10) How many even three digit numbers are there with no repeating digits?
328
Instructor Notes: First take through all 3 digit numbers (900). Then 3 digit numbers with no repeating digits (9.9.8). Hint on the problem (1) Lets start with rightmost digit instead of leftmost. (2) Lets separate the cases where 0 is the rightmost digit versus not. Make sure kids understand the addition of two cases. (BerkeleyMC has a good analysis)
Principal 3: Counting the complement requires subtraction
(BerkeleyMC Chap 2 - Prob 12) Three different flavors of pie are available, and seven children are given a slice of pie so that at least two children get different flavors. How many ways can this be done?
Answer: 3.3.3.3.3.3.3 - 3 = 2184
Instructor Notes: Explain the problem clearly, for example aaaabbc is one possibility. Then calculate number of possibilities without a constraint (3.3.3.3.3.3.3). Then exclude cases. Make sure kids understand why subtraction happens.
(MC Chap 2- Prob 46) We toss the dice three times. How many possible outcomes with at least one occurrence of 6
(6.6.6 - 5.5.5)
Instructor Notes: Ensure kids understand the subtraction
Joke Puzzle: MC Diaries Y1, Chap 3, Warm up 1 - The two sides of a roof slop down at two different angles 60 and 45 degrees. A rooster lays an egg at the very top of the rood - which side will the egg roll down
Answer: Roosters dont lay eggs, hens do
Principal 4: n distinct items can be arranged in n! ways
Instructor Notes: Have kids work with 2, 3, 4 coins of different colors to figure out number of possibilities of arranging them. Drive the insight that number of ways with 4 coins is 4 times the number of ways with 3 coins. Hence introduce the notion and symbol for factorial. Introduce terminology of permutations. Get them to compute factorials for different numbers. Show them how fast it grows.
(Shakuntala - 219) What is the number of batting orders for 9 batsmen?
Answer: 9!
Homework: MartinShCol - 1.14 (colored bowling pins, do with checkers) - there are bowling pins of red color and black color. Can you select 10 pins from these and place them in position of bowling pins, so that no equilateral triangle has all vertices of the same color?
(BerkeleyMC Chap 2 - Prob 8) In how many ways can 10 men and 10 women be paired off
Answer: 10!
Instructor Notes: Its important for kids to understand that it only involves arranging 10 items and not 20. Draw an analogy with just arranging 10 people in a line. Work with smaller numbers as required.
(BerkeleyMC Chap 2 - Prob 9) What if they were not getting married but just making best friends?
Answer: 19.17.15....1
Instructor Notes: Start with smaller numbers. Make sure kids understand how duplicates are being taken care of (for example, always start choosing for remaining people in alphabetic order)
Principal 5: Divide for systematic overcounting
(MC Chap 2- Prob 17) How many words can be formed with letters of word "VECTOR"
Answer: 6!
(MC Chap 2- Prob 18) How many words can be formed with letters of word "TRUST"
Answer: 5!/2
Instructor Notes: Ensure the kids understand the reason for division. For example, take it as T1 and T2, and then show that these are really the same word
(MC Chap 2- Prob 19) How many words can be formed with letters of word "CARAVAN"
Answer: 8!/3!
Instructor Notes: Give kids the time to see that common cases are 3! since there are three As
(MC Chap 2- Prob 20) How many words can be formed with letters of word "MATHEMATICAL"
Answer: 12!/(3! 2! 2!)
(MC Chap 2- Prob 40) Mother has 2 apples, three pears, and four oranges. Every morning she gives one fruit to her son/daughter for breakfast. In how many ways can she do it
Answer: 9!/(2! 3! 4!)
References:
Mathematical Circles (Russian Experience), by Dmitri Fomin, Sergey Genkin, Ilia Itenberg
A Decade of the Berkeley Math Circle. The American Experience, Volume 1. Zvezdelina Stankova, Tom Rike