One hundred ticket numbered \( 1,2,3,...,100 \) are sold to 100 people. There are 4 prizes, including a grand prize of a trip to Australia. How many are there to award the prizes?

• No restrictions.
• People holding 19, 47, 73 and 93 all win prizes.
• The grand prize goes to one of the people holding 19, 47, 73, or 93
• None of the people holding 19, 47, 73 or 93 win prizes.
• The person holding 93 does not win any prize.
• The person holding 93 wins a prize.
• The person holding 93 wins the grand prize.

Select 2 out of 4 \[ r = 2, n = 4 \]

\[ \begin{array}{l} Given\ X = \{a,b,c,d\}:\\ \quad Y = \{\forall x:\\ \qquad\qquad x \in {\cal P}(X):\\ \qquad\qquad\qquad|x| = 2 \}\\ \end{array} \]

• Permutations: All recombinations without repetition \[ ab, ac, ad, ba, bc, bd, ca, cb, cd, da, db, dc \] \[ \left({n!\over (n-r)!}\right) \]

• Combinations: All unique combinations without repetition \[ ab, ac, ad, bc, bd, cd \] \[ \left({n!\over r!\ (n-r)!}\right) \]

\[ \]

Select 2 out of 4 \[ r = 2, n = 4 \]

\[ \begin{array}{l} Given\ X = \{a,b,c,d\}:\\ \quad Y = \{\forall x:\\ \qquad\qquad x \in {\cal P}(X):\\ \qquad\qquad\qquad|x| = 2 \}\\ \end{array} \]

• Permutations: All recombinations with repetition \[ aa, ab, ac, ad, ba, bb, bc, bd, ca, cb, cc, cd, da, db, dc, dd \] \[ \left({n^r}\right)= 4^2 = 16 \]

• Combinations: All unique combinations with repetition \[ aa, ab, ac, ad, bb, bc, bd, cc, cd, dd \] \[ \left({(n+r-1)!\over r!\ (n-1)!}\right) = {(4 + 2 -1)!\over 2! (4-1)!} = {5!\over 2! (3)!} = 5\cdot 2 = 10 \]

No repetitions

• Permutations \[ \left({n! \over (n - r)!}\right) \]

• Combinations \[ \left({n! \over r!\ (n - r)!}\right) \]

Repetition allowed

• Permutations \[ \left(n^r\right) \]

• Combinations: \[ \left({(n+r-1)! \over r!\ (n - 1)!}\right) \]

Sequence is an ordered list of numbers

What would you expect to the be next term?

• 7, 7, 7, 7, 7, …
• 3, −3, 3, −3, 3, …
• 1, 5, 2, 10, 3, 15, …
• 1, 2, 4, 8, 16, 32, …
• 1, 4, 9, 16, 25, 36, …
• 1, 2, 3, 5, 8, 13, 21, …

Find the value for \( n=10 \)

Closed formulas:

• \( a_n = n^2 \)
• \( a_n = {n(n+1)\over 2} \)
• \( a_n = {2^n\over n+2} \)

Recursive formulas:

• \( a_n = 2a_{n-1} \) with \( a_0 = 1 \)
• \( a_n = 2a_{n-1} \) with \( a_0 = 27 \)
• \( a_n = a_{n-1} + a_{n-2} \) with \( a_0 = 0, a_1 = 1 \)

A pair of male/female rabbits are left on a island. The pair does not breed until they are 2 months old. After 2 months old, each pair produces another pair each month. Find the recurrence relation assuming no deaths. -Fibonacci 1202

\[ Pairs_n = Pairs_{n-1} + Pairs_{n-2} \]

• Series: \( n_i = i^2 \)

\[ series: 1, 4, 9, 16, 25, 36, ... \] \[ differ_1: 3, 5, 7, 9, 11, ... \] \[ differ_2: 2, 2, 2, 2, ... \]

• Series: \( n_i = i^2 + 2i + 1 \)

\[ series: 4, 9, 16, 25, 36, 49, 64, 81, ... \] \[ differ_1: 5, 7, 9, 11, 13, 15, 17, ... \] \[ differ_2: 2, 2, 2, 2, 2, 2, ... \]

• Series: \( n_i = i^4 \)

\[ series: 1, 16, 81, 256, 625, 1296, ... \] \[ differ_1: 15, 65, 175, 450, 671, 1105, ... \] \[ differ_2: 50, 110, 194, 302, 434, ... \] \[ differ_3: 60, 84, 108, 132, ... \] \[ differ_4: 24, 24, 24, ... \]

• Series: \( n_i = i^4 + 4 i^3 + 6 i^2 + 4i + 1 \)

\[ series: 16, 82, 258, 628, 1300, 2406, ... \] \[ differ_1: 66, 176, 370, 672, 1106, ... \] \[ differ_2: 110, 194, 302, 434, 590, ... \] \[ differ_3: 84, 108, 132, 156, ... \] \[ differ_4: 24, 24, 24, ... \]

1. Determine the sequence
2. Determine the order of magnitude of the polynomial
3. Determine the polynomial
4. Calculate the number in a stack 15 higher

\[ Balls = a x^3 + b x^2 + c x + d \]

\[ \begin{array}{rcl} 0 &=& d\\ 1 &=& a + b + c \\ 4 &=& 8a + 4b + 2c\\ 10 &=& 27a + 9b + 3c\\ \end{array} \]

\[ \begin{array}{ccccc} 1a & 1b & 1c &=& 1\\ 8a & 4b & 2c &=& 4\\ 27a & 9b & 3c &=& 10\\ \hline 1 & 1 &1&=& 1\\ 0 & 1 &3/2&=& 1\\ 0 & 0 & 1&=& 1/3\\ \hline 1 & 0 & 0 & = & 1/6\\ 0 & 1 & 0 & = & 1/2\\ 0 & 0 & 1 & = & 1/3\\ \end{array} \]

\[ y = {x^3\over 6} + {x^2\over 2} + {x\over 3} \]

\[ y_{15} = 680 \]

Calculate the number of cans in stack 10 levels high

How many students hate potatoes?

• 30 Student choose Mashed, French-fried, or Twice-baked Potatoes
• 15 liked mashed, 20 liked French-fries, and 9 liked twice baked.
• Like 2:
  • 12 students liked both mashed and fried potatoes,
  • 5 liked French fries and twice baked potatoes,
  • 6 liked mashed and baked
• 3 liked all three styles.

• Series: 3,6,12,24,48,96,192

• Differences: 3,6,12,24,48,96,

• Recursive definition:

\[ \begin{array}{rcl} a_0 &=& 3 \\ a_n &=& 2\times a_{n-1}\\ \end{array} \]

• Determining the fixed formula:

\[ \begin{array}{rcl} a_1 &=& 2a_0\\ a_2 &=& 2a_1 = 4a_0\\ a_3 &=& 2a_2 = 8a_0\\ \end{array} \]

• Fixed formula

\[ a_n = 3\times 2^n \]

If it is sunny today, what is the 10 day prediction?

\[ \begin{array}{|l|cc|} \hline & Rain & Sunshine \\ \hline R_n &.6R_n& .4R_n\\ S_n &.5S_n& .5S_n \\ \hline \end{array} \]

\[ \begin{array}{rcl} R_{n+1} &=& 0.6R_n + .5S_n = R_n\\ S_{n+1} &=& 0.4R_n + .5S_n = S_n\\ 1 &=& R + S\\ \end{array} \]

Solution: \[ R = 5/9; S= 4/9 \]

• Tally of level changes

Equalibrium Polution

• Good (1) : 00.0%
• Moderate (2) : 40.0%
• High (3) : 38.4%
• Very high 4) : 18.5%
• Hazardous (5) : 10.7%
• Extreme (6) : 2.1%

12=>{3=>1.00},
22=>{2=>0.652,3=>0.348},
23=>{2=>0.231,3=>0.461,4=>0.308},
32=>{2=>0.625,3=>0.375},
33=>{2=>0.235,3=>0.471,4=>0.235,6=>0.059},
34=>{2=>0.2,3=>0.3,4=>0.4,6=>0.1},
36=>{5=>1.0},
42=>{2=>0.75,3=>0.25},
43=>{2=>0.25,3=>0.5,4=>0.25},
44=>{2=>0.4,3=>0.2,4=>0.4},
46=>{5=>1.0},
53=>{3=>1.00},
55=>{3=>1.00},
65=>{3=>0.5,5=>0.5}}}

\[ \tiny\begin{array}{lcccccc} State & 1st & 2nd & 3rd & 4th & Grd & With & Retir \\ \hline 1st & 0.1 & 0.7 & 0.0 & 0.0 & 0.0 & 0.2&0.0 \\ 2nd & 0.1 & 0.8 & 0.0 & 0.0 & 0.1 &0.0 & 0.0 \\ 3rd & 0.0 & 0.0 & 0.1 & 0.8 & 0.0 & 0.1 &0.0 \\ 4th & 0.0 & 0.0 & 0.0 & 0.1 & 0.8 & 0.1 &0.0 \\ Grad & 0.0& 0.0 & 0.0 & 0.0 & 1.0 & 0.0& 0.0 \\ With & 0.001&0.0& 0.0 & 0.0 & 0.0 &0.009& 0.99\\ Retire & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 &1.0\\ \hline \end{array} \]

• Comparison between binary run lengths in 5 random population and the experimental data