1. Summary
What have we learned?

2. Coin Toss Combine group experimental results Theoretical results40
limit/or extra tosses to have exactly 96 30
Theoretical results
mult. by? 20
0H, 5T ___ ___
1H, 4T ___ ___
2H, 3T ___ ___
3H, 2T ___ ___
4H, 1T ___ ___
5H, 0T ___ ___
Total ___ ___ 10 96
0H 1H 2H 3H 4H 5H
How do the theoretical and experimental results compare?

3. Pascal’s Triangle 1
nC0 nC1 nC nCr nCn-1 nCn
row 0
row 1
row 2
row 3
row 4
row 5 . .
row n
nCr “n choose r”
the number of ways you can get r objects when selecting from n objects
“object” can be a thing or an event

4. Binomial Theorem (x + y) two objects, x and y, one time
compare to H or T
(x + y)2 = (x + y)(x + y) two objects, x and y, two times
when multiply by x and then by y, end up with all possible combinations of the two variables multiplied together
compare to HH,HT,TH, TT
(x + y)3 = (x + y)(x + y)(x + y) two objects, three times
all possible combinations of the two variables multiplied together three times
compare to ,HHH,HHT,HTH,HTT,THH,THT,TTH, TTT
Pascal’s Triangle gives the coefficients (how many times each group occurs), then append the variables to each term
(x + y)n = nC0xn + nC1xn-1y + nC2xn-2y
+ nCrxn-ryr nCn-1x1yn-1 + nCnyn
x exponent is number of x’s, y exponent is r, the number of y’s sum of the exponents always equals n

5. n(n-1)(n-2)…2·1 = n! Permutations of n objects
nCr
How many ways can we order n objects?
n(n-1)(n-2)…2·1 = n! Permutations of n objects
What if we are interested in r of the n objects? (but order still matters)
1st, 2nd, and 3rd finishers in a race with 9 contestants 9P3 = 9·8·7
n(n-1)(n-2)…2· n! Permutations of n objects taken r at a time
(n-r)…2· (n-r)! nPr
ways we can have the “don’t care” combinations, the ordering of the n-r remaining objects
What if we are interested in r of the n objects? (and order doesn’t matter)
n(n-1)(n-2)…2· n! Combinations of n objects taken r at a time (n-r)…2·1(r)(r-1)… 2·1 (n-r)!r! nCr =
r!, the number of ways we can order the r objects