Counting

  1. There are 10 CS majors and 20 SE majors.

    • How many ways are there to pick 4 representatives, two from CS students and 2 from SE students?

    • How many ways are there to pick 4 representatives from all CS and SE students?

  2. A new 10-storey building has 30 windows on each floor. How many windows are in the building?

  3. A factory make polo shirts in 12 colors, in male and female versions, and 3 sizes for each sex. How many different types of shirts are made at this factory?

  4. There are 2 flights from Mandalay to Chiang Mai and 9 flights from Chiang Mai to Bangkok. How many ways are there to fly from Mandalay to Bangkok via Chiang Mai?

  5. How many unique license plates can be created using 2 of the 44 Thai characters and 3 numeric digits?

  6. A drawer has 12 socks: 4 are red and 8 are blue. You need to select socks from this drawer in complete darkness.

    • How many socks do you need to select before you can be certain you have at least 2 blue socks?
    • How many socks do you need to select before you can be certain that you have at least 2 red socks?
    • How many socks do you need to select before you can be certain that you have at least 1 red and 1 blue sock?

Permutations

  1. How many bit strings can be made from 5 binary digits?

  2. How many different wedding photgraphs of six people (including the bride and groom) can be taken where:

    • the bride stands next to the groom

    • the bride stands on the left of the groom

    • the bride does not stand next to the groom

  3. How many kinds of pancakes can be made from 3 types of flour, 3 types of sugar and 2 types of milk?

  4. How many types of subway sandwiches can be made from 5 different kinds of bread, 4 different kinds of sliced meat, 3 different sauces and any selection of 3 vegetables from a range of 10?

  5. 20 Students were asked about ice cream flavors. 14 Liked Strawberry, 12 liked chocolate and 10 liked vanilla. Within the same group, 8 liked both strawberry and chocolate, 6 liked both vanilla and chocolate and 7 liked both strawberry and vanilla. There were also 5 who replied that they liked all 3 flavors. Provide the following numbers:
    • Number who only like strawberry.
    • Number who only like chocolate.
    • Number who only like vanilla.
    • Number who do not like strawberry, chocolate or vanilla.

asked about all 3 # Combinations

  1. How many way scan 5 pens be given to 3 students?

  2. How many six letter words could be formed from the word “teethe”?

  3. If you have this map for the paths between A and B, how many paths exist between A and B if one can only go EAST OR North?

\[\begin{array}{c|c|c|c|c|c} & & & & & B \\ \hline & & & & & \\ \hline & & & & & \\ \hline A& & & & & \\ \end{array} \]

  1. How many different kinds of 2-scoop ice cream cones can be made at Baskin Robins which has 31 flavors of ice cream and 3 different types of cone?

Series

  1. For each in the following series:

    • \(10 ,17 ,24 ,31 ,38 ,45 ,52 ,59 ,66\)
    • \(6 ,7 ,10 ,15 ,22 ,31 ,42 ,55 ,70\)
    • \(9 ,11 ,13 ,15 ,17 ,19 ,21 ,23 ,25\)
    • \(3 ,12 ,27 ,48 ,75 ,108 ,147 ,192 ,243\)
    Determine the following:
    • The next expected value in the series
    • The recursive definition of the elements
    • The direct formula for the elements

  2. What is the order of the polynomial that defines these series:

    • \(1 ,20 ,75 ,184 ,365 ,636 ,1015 ,1520 ,2169\)

    • \(1 ,15 ,79 ,253 ,621 ,1291 ,2395 ,4089 ,6553\)

    • \(1 ,3 ,7 ,13 ,21 ,31 ,43 ,57 ,73\)

Summation

  1. Calculate the sum of this series of numbers:

    • \(5 + 10 + 15 + ... + 100 =\)

    • \(1 + 2 + 4 + 8 + ... + 512 =\)

  2. If on 1 Mar you add 1 baht to your bank account, 2 baht on 2 Mar , 3 baht on 3 Mar and continue like this to the end of the month, how much will you have in the bank on 31 Mar?

  3. The author of TeX gives a reward for every bug found according to this formula: \(Reward_n = 2^{(n-1)}\). So far only 9 bugs have been found.

    • How much money has the author paid out in rewards?
    • How much is the reward for the next bug found?

Solutions:

Counting

  1. There are 10 CS majors and 20 SE majors.

  • 2 representatives from CS students and 2 from SE:

    \[\left({10 \atop 2}\right)\left({20 \atop 2}\right) = {10!\ 20!\over 2! 8! 2! 18!} = 5 \times 9 \times 10 \times 19 = 8550\]

  • 4 representatives from all students:

    \[\left({30 \atop4}\right) = {30! \over 4! 26!} = {30 \times 29 \times 28 \times 27\over 4!} = 27,405\]

  1. 10-storey building

\[ 10 \times 30 = 300\]

  1. Shirt factory:

\[12 \times 2 \times 3 = 72\]

  1. Flights from Mandalay to Bangkok via Chiang Mai: \[2 \times 9 = 18\]

  2. Thai license plates:

\[ 44 \times 44 \times 10 \times 10 \times 10 = 1,936,000\]

  1. Sock drawer:

    • at least 2 blue socks: (All the red plus 2) \[4 + 2 = 6\]

    • least 2 red socks: (All the blue plus 2) \[8+ 2= 10\]

    • least 1 red and 1 blue sock: (all red plus one) \[4 + 1= 5\]

Permutations

  1. How many bit strings can be made from 5 binary digits?

\[2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32\]

  1. Wedding photgraphs:

    • the bride stands next to the groom: \[Photo_r = \left\{[BG],p1,p2,p3,p4\right\}\] \[Photo_l = \left\{[GB],p1,p2,p3,p4\right\}\]

    \[2 \times 5! = 240\]

    • the bride stands on the left of the groom \[5! = 120\]

    • the bride does not stand next to the groom (Calculate for each possible combination of bride and groom positions)

\[\begin{array}{c} B4321G + B432G1 + B43G21 + B4G321 + \\ 4B321G + 4B32G1 + 4B3G21 + G4B321 + \\ 43B21G + 43B2G1 + 4G3B21 + G43B21 + \\ 432B1G + 43G2B1 + 4G32B1 + G432B1 + \\ 432G1B + 43G21B + 4G321B + G4321B \\ \end{array}\]

\[ 5 \times 4 \times 4! = 480\]

  1. Pancakes:

\[3 \times 3 \times 2= 18\]

  1. Subway sandwiches:

\[5 \times 4 \times 3 \right({10 \atop 3}\right) = {60 \times 10!\over 3! 7!} = 10 \times 10 \times 9 \times 8 = 7200\]

  1. Ice cream: Venn diagram
  • only like strawberry: 4
  • only like chocolate: 3
  • only like vanilla: 2
  • do not like strawberry, chocolate or vanilla: 0

Combinations

  1. How many way can 5 pens be given to 3 students?

\[\left( {(n+r-1)!\over r!(n - 1)! } \right)={(5+3-1)! \over 5! 2!} = {7\times 6 \over 2} = 7 \times 3 = 21\]

\[\begin{array}{|c|c|} \hline *****|-|- & *|****|- \\ & *|***|* \\ ****|*|- & *|**|**\\ ****|-|* & *|*|***\\ & *|-|****\\ ***|**|- & \\ ***|*|* & -|*****|- \\ ***|-|** & -|****|*\\ & -|***|** \\ **|***|- & -|**|*** \\ **|**|* & -|*|**** \\ **|*|** & -|-|***** \\ **|-|*** & \\ \hline \end{array}\]

  1. Six letter words from the word “teethe”?

\[{6! \over 3! 2! 1!} = {6 \times 5 \times 4\over2} = 60\]

eeehtt, eeetht, eeetth, eehett, eehtet, eehtte, eeteht, eeteth, eethet, eethte, eetteh, eetthe, eheett, ehetet, ehette, ehteet, ehtete, ehttee, eteeht, eteeth, etehet, etehte, eteteh, etethe, etheet, ethete, ethtee, etteeh, ettehe, etthee, heeett, heetet, heette, heteet, hetete, hettee, hteeet, hteete, htetee, htteee, teeeht, teeeth, teehet, teehte, teeteh, teethe, teheet, tehete, tehtee, teteeh, tetehe, tethee, theeet, theete, thetee, thteee, tteeeh, tteehe, ttehee, ttheee

  1. If you have this map for the paths between A and B, how many paths exist between A and B if one can only go EAST OR North?

\[\begin{array}{c|c|c|c|c|c} & & & & & B \\ \hline & & & & & \\ \hline & & & & & \\ \hline A& & & & & \\ \end{array} \]

\[\left( {6 \atop 2}\right) = {6!\over 2! 4!} = {6 \times 5\over 2} = 15\]

\[\begin{array}{|c|c|c|} \hline EEEENN & EENNEE & NENEEE \\ EEENEN & ENENEE & NEENEE \\ EEENNE & ENEENE & NEEENE \\ EENENE & ENEEEN & NEEEEN \\ EENEEN & ENNEEE & NNEEEE \\ \hline \end{array}\]

  1. How many different kinds of 2-scoop ice cream cones can be made at Baskin Robins which has 31 flavors of ice cream and 3 different types of cone?

\[\left({31 \atop 2}\right) \times 3 = {3 \times 31!\over 2! 29!} = {3 \times 31 \times 30 \over 2} = 93 \times 15 = 1395\]

Series

  1. For each in the following series:
  • \(10 ,17 ,24 ,31 ,38 ,45 ,52 ,59 ,66\)

\[\begin{array}{c} 10 ,17 ,24 ,31 ,38 ,45 ,52 ,59 ,66\\ 7,7,7,7,7,7,7,7\\ \end{array}\]

  • The next expected value in the series: 73

  • The recursive definition of the elements \[x_n = x_{n-1} + 7\]

  • The direct formula for the elements \[x_n = 3 + 7n\]

  • \(6 ,7 ,10 ,15 ,22 ,31 ,42 ,55 ,70\) \[\begin{array}{c} 6 ,7 ,10 ,15 ,22 ,31 ,42 ,55 ,70\\ 1 , 3, 5 , 7 ,9 ,11 ,13, 15\\ 2,2,2,2,2,2,2\\ \end{array}\]

    • The next expected value in the series: 87

    • The recursive definition of the elements \[x_n = 2x_{n-1}+x_{n-2}+ 2\]

    • The direct formula for the elements

\[\begin{array}{c} y = an^2 + bn + c\\ 6 = a + b + c\\ 7 = 4a + 2b + c\\ 10 = 9a + 3b + c\\ \hline 1 = 3a + b\\ 4 = 8a + 2b\\ \hline 2 = 6a +2b\\ 4 = 8a + 2b\\ \hline 2 = 2a\\ \hline a = 1; b = -2; c = 7\\ x_n = n^2 -2n + 7\\ \end{array}\]

  • \(9 ,11 ,13 ,15 ,17 ,19 ,21 ,23 ,25\)

\[\begin{array}{c} 9 ,11 ,13 ,15 ,17 ,19 ,21 ,23 ,25\\ 2,2,2,2,2,2,2,2\\ \end{array}\]

  • The next expected value in the series: 27

  • The recursive definition of the elements \[x_n = x_(n-1) + 2\]

  • The direct formula for the elements \[x_n = 9 + 2(n-1)\] —-

  • \(3 ,12 ,27 ,48 ,75 ,108 ,147 ,192 ,243\) \[\begin{array}{c} 3 ,12 ,27 ,48 ,75 ,108 ,147 ,192 ,243\\ 9, 15, 21, 27, 33, 39, 45, 51\\ 6, 6, 6 ,6 ,6 ,6 ,6\\ \end{array}\]

  • The next expected value in the series: 300

  • The recursive definition of the elements: \[x_n = 2x_{n-1} - x_{n-2} + 6\]

  • The direct formula for the elements

\[\begin{array}{c} y = an^2 + bn + c\\ \hline 3 = a + b + c\\ 12 = 4a +2b + c\\ 27 = 9a +3b +c\\ \hline 9 = 3a+b\\ 24 = 8a+2b\\ \hline 18 = 6a + 2b\\ 24 = 8a + 2b\\ \hline 6 = 2a\\ a = 3; b = 0; c = 0\\ x_n = 3n^2\\ \end{array}\]

  1. The order of the polynomials:
  • \(1 ,20 ,75 ,184 ,365 ,636 ,1015 ,1520 ,2169\)

\[\begin{array}{rc} 0 & 1,20,75,184,365,636,1015,1520,2169\\ 1 & 19 55, 109, 181, 271, 379, 505, 649\\ 2 & 36 ,54 ,72 ,90 ,108 ,126, 144\\ 3 & 18 ,18 ,18 ,18 ,18 ,18\\ \end{array}\]

  • \(1 ,15 ,79 ,253 ,621 ,1291 ,2395 ,4089 ,6553\)

\[\begin{array}{rc} 0& 1 ,15 ,79 ,253 ,621 ,1291 ,2395 ,4089 ,6553\\ 1 & 14 , 64, 174 ,368 , 670 ,1104 ,1694 ,2464\\ 2 & 50 ,110 ,194 ,302 ,434 ,590 ,770\\ 3 & 60, 84, 108, 132, 156, 180\\ 4& 24 ,24 ,24 ,24, 24\\ \end{array}\]

  • \(1 ,3 ,7 ,13 ,21 ,31 ,43 ,57 ,73\)

    \[\begin{array}{rc} 0&1 ,3 ,7 ,13 ,21 ,31 ,43 ,57 ,73\\ 1 & 2 ,4 , 6, 8, 10 ,12 ,14, 16\\ 2 & 2, 2, 2 ,2 ,2 ,2 ,2\\ \end{array}\]

Summation

  1. Calculate the sum of this series of numbers:

  • \(5 + 10 + 15 + ... + 100 =\)

    \[\left({5+100\over 2}\right)\left({100\over 5}\right)={105 \times 100\over 10}= 1050\]

  • \(1 + 2 + 4 + 8 + ... + 512 =\)

\[\left({ar^{n+1}-a\over r-1}\right)={2^{10} - 1\over 2-1} = {1024-1\over 1} = 1023\]

  1. Bank account:

\[1+2+3+...+31 = 31\left({1+31\over 2}\right)= 31 \times 16=496\]

  1. TeX Reward:

    • How much money has the author paid out in rewards? \[{2^{9} -1\over 2-1}={512 -1\over 1}=511\]

    • How much is the reward for the next bug found? \[Reward_{10} = 2^{(n-1)}= 2^{(10-1)} = 2^9 = 512\]