Pascal s riangle: Flipping Coins & Binomial Coefficients Robert Campbell 5/6/204
An Experiment Coin Flips Break into teams Flip a coin 6 times Count the number of heads Do this 64 times per team Graph the results ow many times did each number of heads occur? 5/6/204 2
Graphing Coin Flips 20 5 5 0 6 6 2 3 4 5 6 5/6/204 3
Ways to Get or 2 eads Flip 4 coins: All eads can only be gotten one way: 3 eads (and ail) can be gotten four ways: 2 eads (and 2 ails) can be gotten six ways: 5/6/204 4
Coins & Counting I Given Five pennies Four eads and one ails ow many ways can I arrange them in a line? 5/6/204 5
Coins & Counting II Given Five pennies hree heads and two tails ow many ways can I arrange them in a line? 5/6/204 6
Coins & Counting III 2 heads and 3 tails (Same as 3 heads and 2 tails) head and 4 tails (Same as 4 heads and tail) No heads and 5 tails Only one way (Same as 5 heads and no tails) (5P: 0, 5) = (5P:, 4) = 5 (5P: 2, 3) = 0 (5P: 3, 2) = 0 (5P: 4, ) = 5 (5P: 5, 0) = 5/6/204 7
5/6/204 8 Coins & Counting IV Six pennies: 4 heads, 2 tails 5 5 0 2 5 5 2 5 2 6 So
5/6/204 9 Counting Coin Flips
he Arithmetic riangle +=2 +2=3 +2=3 +3=4 3+3=63 +3=4 +4=5 4+6=0 4+6=0 +4=5 5/6/204 0
he Arithmetic riangle he Rules: Make a triangle out of numbered boxes box in first row, 2 boxes in second row, Leftmost and rightmost boxes in each row are Other boxes - Add the two boxes above them 2 3 3 4 6 4 5 0 0 5 6 5 20 5 6 5/6/204
riangle s istory Chinese: Jia Xian [00-070] Yang ui [238-298] Indian Arabic Pingala [200 BC?] - Meru-prastara (staircase to Mt Meru) Al-Karaji [953-029] Omar Khayyam [048-3] European artaglia [499-557, Italy] Pascal [623-662, France] 5/6/204 2
Properties I he Sums st Row - 2 nd Row + = 2 3 rd Row +2+ = 4 4 th Row +3+3+ = 8 5 th Row +4+6+4+ = 6 6 th Row +5+0+0+5+ = 32 Next row sum is twice as we add each box twice 5/6/204 3
Properties II he Diagonals st Diag All s 2 nd Diag he integers 3 rd Diag riangular numbers 4 th Diag etrahedral numbers 5/6/204 4
Coloring the riangle Color the odd squares black, the even squares white Do you need to actually add? Even + Even = Even Odd + Even = Odd Odd + Odd = Even Color the triangle mod 3 If 3 divides the number - Black If 3 divides the number plus Red If 3 divides the number plus 2 White Why are there triangles in the pattern? 2 3 3 4 6 4 5 0 0 5 6 5 20 5 6 5/6/204 5
Singmaster s Conjecture Note: Any number n not not found below nth row So, n only occurs finitely often, say #n Conj: For some N, #n<n for all n Examples: 2 occurs once (all others occur more often) 3, 4, 5 occur twice 6 occurs three times Occurring 6 times: 20, 20, 540, Occurring 8 times: 3003, Does any number occur 5, 7 or more than 8 times? 5/6/204 6
References & Resources Pascal s Arithmetical riangle, A. W. F. Edwards, 2002 Pascal s riangle, V. A. Uspenskii, 974 Slides, Coin Flips & Coloring the riangle http://userpages.umbc.edu/~rcampbel/mepp/pascal-riangle/ Math Forum (class materials) http://mathforum.org/workshops/usi/pascal/ Wikipedia http://en.wikipedia.org/wiki/pascal s_triangle 5/6/204 7
Backup Slides 5/6/204 8
Chevalier de Méré s Bets Bet: in four dice rolls it will come up 6 at least once. Bet: in 24 rolls of two dice we will see double-6 at least once. So, my friend Monsieur Pascal, why do I usually win the first bet but lose the second? A friend and I were playing, having agreed that the first to win 0 rounds won the pot. We had to quit early when I had won 7 rounds and he had won 5. ow should we split the pot? 5/6/204 9
Factorials I x4 x3 x2 x = 24 ow many ways can I line up 4 people? Choose one person to stand on the left 4 possible choices Choose one more to stand next to him 3 people to choose from Choose one more to stand next to him 2 people left to choose from Choose the last one No real choice - only one is left (4)(3)(2)() = 24 Possible ways 5/6/204 20
Factorials II (4)(3)(2)() is called Four Factorial and is written 4! wo factorial is 2! = (2)() = 2 hree factorial is 3! = (3)(2)() = 6 Four factorial is 4! = (4)(3)(2)() = 24 en factorial is 0! = (0)(9)(8)(7)(6)(5)(4)(3)(2)() = 3628800 5/6/204 2
Binomial Coefficients I he numbers in Pascal s riangle are called: Binomial Coefficients Choose numbers he number of ways of lining up 5 pennies, 3 heads, 2 tails, is called (5 choose 2) or (5 choose 3) (Lining up 5 pennies and choosing 2 of them to be heads) (5 choose 0) = (5 choose ) = 5 (5 choose 2) = 0 (5 choose 3) = 0 (5 choose 4) = 5 (5 choose 5) = 5/6/204 22
Binomial Coefficients II o compute the Choose Number (6 choose 3): Fill in the first 6 rows of Pascal s riangle OR - compute (6 choose 3) directly ow many ways can I choose 3 out of 6 possibilities? Choose one (6 possible choices) Choose another (5 possible choices) Choose a third (4 possible choices) So (6)(5)(4) = 20 possible choices But I don t care which one I chose first - what order I chose So ow may ways can I rearrange the 3 I did choose? here are (6)(5)(4)/3! = 20 ways to choose 3 out of 6 5/6/204 23
Binomial Coefficients III (6 choose 3) = (6)(5)(4)/3! = (6)(5)(4)(3)(2)()/((3)(2)()3!) = 6!/(3! 3!) Compute: (6)(5)(4)/((3)(2)()) = (6/3)(5)(4/2) = 20 (0 choose 3) = (0)(9)(8)/3! = (0)(9)(8)(7)(6)(5)(4)(3)(2)()/(7)(6)(5)(4)(3)(2)() 3!) = 0!/(7! 3!) Compute: (0)(9)(8)/((3)(2)()) = (0)(9/3)(8/2) = (0)(3)(4) = 20 (2 choose 4) = (2)()(0)(9)/4! = 2!/(8! 4!) Compute: (2)()(0)(9)/((4)(3)(2)()) = (2/4)()(0/2)(9/3) = (3)()(5)(3) = 495 5/6/204 24
More Graphing Coin Flips 5 20 5 Bell Curve? Central Limit heorem ry 2 20 = 048576 trials of 20 flips each 6 6 0 2 3 4 5 6 5/6/204 25
Another Experiment - Coloring Demonstrate by coloring 8-triangle even-odd Demonstrate by coloring 8-triangle mod 3 ry mods 3-0 Only those 0 mod N Separate color for each mod 5/6/204 26
Multinomials I ow many ways can I place 4 poker chips in a line if 2 are red, is white and is blue (and I don t care which red chip is which)? ow many ways can I arrange (4Chips; 4red, 0white, 0blue)? ow many ways can I arrange (4Chips; 3red, white, 0blue)? ow many ways can I arrange (4Chips; 2red, 2white, 0blue)? ow many ways can I arrange (4Chips; 2red, white, blue)? ow many ways can I arrange 6 chips if 3 are red, 2 white and blue? 5/6/204 27
Multinomials II he number of ways to arrange objects of more than two types are multinomial coefficients. here are (6; 3, 2, ) = 6!/(3! 2!!) = 60 ways to arrange 6 chips if 3 are red, 2 white and blue. he multinomials fit together in a pyramid, just like the binomials fit together in a triangle. 5/6/204 28
5/6/204 29 Combinations Flip a coin N times ow many ways are there to get heads?!!! N N N 2 3 4 2 3 4 0!0! 4 4! 0 4 4 2 3 2 3 4!! 4 4! 4 6 4 24 2 2 2 3 4 2!2! 4 4! 2 4 Flip a coin 4 times here is way to get no heads here are 4 ways to get head here are 6 ways to get 2 heads Flip a coin 8 times here are 56 ways to get 3 heads 56 720 40320 2 3 2 3 4 5 2 3 4 5 6 7 8 3!3! 8 8! 3 8
Polynomials & Binomial Coeffs (x + y) is called a binomial because it has two terms (x + y) 2 = x 2 + 2xy + y 2. (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3. (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4. (x + y) 5 = x 5 + 5x 4 y + 0x 3 y 2 + 0x 2 y 3 + 5xy 4 + y 5. he binomial coefficients are called that because they are coefficients of powers of binomials. 5/6/204 30
Early istory of the riangle Jia Xian [ th C, China] Bhaskara [2 th C, India] Omar Khayyam [2 th C, Persia] Zhu Shijie [4 th C, China] Levi ben Gershon [4 th C, France] artaglia [6 th C, Italy] Pascal [7 th C, France] many others 5/6/204 3
An Exercise Fill the riangle Given a blank form with rows through 7 (optional could easily do on any blank sheet of paper) Fill in the entries of the triangle Save for later use, looking for patterns 5/6/204 32
he Arithmetic riangle +=2 +2=3 +2=3 +3=4 3+3=63 +3=4 +4=5 4+6=0 4+6=0 +4=5 5/6/204 33