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#set math.mat(delim: "[")
#set page("us-letter")
#let solve(solution) = {
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    fill: rgb(0, 149, 255, 15%),
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#align(center)[
  = CS 3333 Mathematical Foundations
  Homework 5 (100 pts)\
  #underline[Price Hiller] | #underline[zfp106]
]
#line(length: 100%, stroke: .25pt)


*Questions*\
Please write down the major intermediate steps.

1. Calculate the rank of the following matrices. (12 pts)

  #grid(
    columns: 2,
    align: center,
    gutter: 2em,
    [
      $A = mat(
        2, -1, 3;
        1, 0, 1;
        0, 2, -1;
        1, 1, 4;
      )$
    ],
    [
      (6 pts)
    ],
    grid.cell(
      colspan: 2,
      [
        #note[
          #grid(
            column-gutter: (1em, 4em, 1em, 1em),
            columns: 2,
            align: left + horizon,
            row-gutter: 2em,
            [$R_4 = R_4 - R_2$],
            [$mat(
                2, -1, 3;
                1, 0, 1;
                0, 2, -1;
                0, 1, 3;
              )$],
            [$R_1 = 1 / 2 R_1$],
            [$mat(
                1, -1/2, 3/2;
                1, 0, 1;
                0, 2, -1;
                0, 1, 3;
              )$],

            [$R_2 = R_2 - R_1$],
            [$mat(
                1, -1/2, 3/2;
                0, 1/2, -1/2;
                0, 2, -1;
                0, 1, 3;
              )$],
            [$R_2 = 2R_2$],
            [$mat(
                1, -1/2, 3/2;
                0, 1, -1;
                0, 2, -1;
                0, 1, 3;
              )$],

            [$R_4 = R_4 - R_2$],
            [$mat(
                1, -1/2, 3/2;
                0, 1, -1;
                0, 2, -1;
                0, 0, 4;
              )$],
            [$R_3 = R_3 - 2R_2$],
            [$mat(
                1, -1/2, 3/2;
                0, 1, -1;
                0, 0, 1;
                0, 0, 4;
              )$],

            [$R_4 = R_4 - 3R_3$],
            [$mat(
                1, -1/2, 3/2;
                0, 1, -1;
                0, 0, 1;
                0, 0, 0;
              )$],
            grid.cell(colspan: 2, align: center, solve[The rank of $A$ is 3.])
          )
        ]
      ],
    ),
    colbreak(),
    colbreak(),
    [
      $B = mat(
        3, 2, -1;
        2, -3, -5;
        -1, -4, -3;
      )$
    ],
    [
      (6 pts)
    ],
    grid.cell(
      colspan: 2,
      [
        #note[
          #grid(
            column-gutter: (1em, 4em, 1em, 1em),
            columns: 2,
            align: left + horizon,
            row-gutter: 2em,
            [$R_2 = R_2 + 2R_4$],
            [$mat(
              3, 2, -1;
              0, -11, -11;
              -1, -4, -3;
            )$],

            [$R_2 = -1 / 11 R_2$],
            [$mat(
              3, 2, -1;
              0, 1, 1;
              -1, -4, -3;
            )$],

            [$R_4 ↔ R_2$],
            [$mat(
              3, 2, -1;
              -1, -4, -3;
              0, 1, 1;
            )$],

            [$R_2 = R_2 + 1 / 3R_1$],
            [$mat(
              3, 2, -1;
              0, -10/3, 10/3;
              0, 1, 1;
            )$],

            [$R_2 = -3 / 10 R_2$],
            [$mat(
              3, 2, -1;
              0, 1, 1;
              0, 1, 1;
            )$],

            [$R_3 = R_3 - R_2$],
            [$mat(
              3, 2, -1;
              0, 1, 1;
              0, 0, 0;
            )$],
            grid.cell(colspan: 2, align: center, solve[The rank of $B$ is 2.])
          )
        ]
      ],
    ),
  )




2. Solve the following systems using the inverse of a matrix. (12 pts)

  #grid(
    columns: 2,
    align: center,
    gutter: 2em,
    [
      $
        -7x + 3y &= -34\
        8x -4y &= 44
      $
    ],
    [
      (4 pts)
    ],
    grid.cell(
      colspan: 2,
      [
        #note[
          #columns(2)[
            $
              cases(
                -7x + 3y = -34\
                8x -4y = 44
              ) -> mat( -7, 3, -34; 8, -4, 44; augment: #2)\
            $
            $
              R_1 &= -1 / 7 R_1& &-> mat( 1, -3/7, -34/7; 8, -4, 44; augment: #2)\
              R_2 &= R_2 - 8R_1& &-> mat( 1, -3/7, -34/7; 0, -4/7, 36/7; augment: #2)\
              R_2 &= -7 / 4R_2& &-> mat( 1, -3/7, -34/7; 0, 1, -9; augment: #2)\
              R_1 &= R_1 + 3 / 7R_2& &-> mat(1, 0, 1; 0, 1, -9; augment: #2)\
            $
            #solve[$x = 1, y = -9$]
          ]
        ]
      ],
    ),
    [

      $
        5x + 15y + 56z &= 35\
        -4x - 11y -41z &= -26\
        -x - 3y - 11z &= -7
      $
    ],
    [
      (8 pts)
    ],
    grid.cell(
      colspan: 2,
      [
        #note[
          #block(inset: (left: 30pt, right: 30pt))[
            $
              cases(
                5x + 15y + 56z& =& 35\
                -4x - 11y -41z& =& -26\
                -x - 3y - 11z& =& -7
              ) -> mat(5, 15, 56, 35; -4,-11,-41,-26;-1,-3,-11,-7; augment: #3)\
            $
            $
              R_1 &= 1 / 5 R_1& &-> mat(1, 3, 56/5, 7; -4,-11,-41,-26;-1,-3,-11,-7; augment: #3)\
              R_2 &= R_2 + 4R_1& &-> mat(1, 3, 56/5, 7; 0, 1, 19/5, 2;-1,-3,-11,-7; augment: #3)\
              R_3 &= R_3 + R_1& &-> mat(1, 3, 56/5, 7; 0, 1, 19/5, 2;0,0,1/5,0; augment: #3)\
              R_1 &= R_1 - 3R_2& &-> mat(1,0,-1/5,1; 0, 1, 19/5, 2;0,0,1/5,0; augment: #3)\
              R_3 &= 5R_3& &-> mat(1,0,-1/5,1; 0, 1, 19/5, 2;0,0,1,0; augment: #3)\
              R_1 &= R_1 +1 / 5 R_3& &-> mat(1,0,0,1; 0, 1, 19/5, 2;0,0,1,0; augment: #3)\
              R_2 &= R_2 - 19 / 5 R_3& &-> mat(1,0,0,1; 0, 1, 0, 2;0,0,1,0; augment: #3)\
            $
            #solve[$x = 1, y = 2, z=0$]
          ]
        ]
      ],
    ),
  )

#align(center + bottom, note[#text(size: 3em)[⇊ SEE NEXT PAGE ⇊]])
#pagebreak()

3. Find the eigenvalues and eigenvectors of the following matrices. (12 pts)

  #grid(
    columns: 2,
    align: center,
    gutter: 2em,
    [
      $A = mat(
        2, 3;
        -3, -5;
      )$
    ],
    [
      (4 pts)
    ],
    grid.cell(
      colspan: 2,
      [
        #note[
          #block(inset: (left: 30pt, right: 30pt))[
            #underline[Calculating Eigenvalues]
            $
              A - λ ⋅"I" &= mat(2 - λ, 3;-3, -5 - λ)&\
              mat(delim: "|", 2 - λ, 3;-3, -5 - λ) &= 0&\
              (2 - λ)(-5-λ) - 3(-3) &=0&\
              (2 - λ)(-5-λ) + 9 &=0&\
              (-3 + sqrt(13)) / 2,(-3 - sqrt(13)) / 2&= λ&\
            $
            #underline[Calculating Eigenvectors]
            #solve[
              Eigen Values: $(-3 + sqrt(13)) / 2,(-3 - sqrt(13)) / 2$
            ]
          ]
        ]
        #align(left)[#text(
            red,
            weight: "black",
          )[And it is at this point that I have run out of time :(. I forgot this was due tonight, I thought today was the 13th until I saw that football was on at 9pm. My stomach dropped a few miles as you might imagine. At this point, I've winged it and scraped what I could together. May the dice roll in my favor.]]
      ],
    ),
    [
      $B = mat(
        2, 0, 0;
        0, 4, 5;
        0, 4, 3;
      )$
    ],
    [
      (8 pts)
    ],
    grid.cell(colspan: 2, []),
  )


4. List all the permutations of ${P,Q, R, S}$ (6 pts).

  #solve[
    #grid(
      align: center,
      columns: 4,
      [
        ${P,Q,R,S}$
        ${P,Q,S,R}$
        ${P,R,Q,S}$
        ${P,R,S,Q}$
        ${P,S,Q,R}$
        ${P,S,R,Q}$
      ],
      [
        ${Q,P,R,S}$
        ${Q,P,S,R}$
        ${Q,R,P,S}$
        ${Q,R,S,P}$
        ${Q,S,P,R}$
        ${Q,S,R,P}$
      ],
      [
        ${R,P,Q,S}$
        ${R,P,S,Q}$
        ${R,Q,P,S}$
        ${R,Q,S,P}$
        ${R,S,P,Q}$
        ${R,S,Q,P}$
      ],
      [
        ${S,P,Q,R}$
        ${S,P,R,Q}$
        ${S,Q,P,R}$
        ${S,Q,R,P}$
        ${S,R,P,Q}$
        ${S,R,Q,P}$
      ],
    )]

#align(center + bottom, note[#text(size: 3em)[⇊ SEE NEXT PAGE ⇊]])
#pagebreak()

5. Calculate the value of each of these quantities (10 pts).

  #grid(
    columns: 3,
    align: center,
    gutter: 2em,
    [
      a. $P(7,2)$
      #solve[$7! / (7 - 2)! = 42$]
    ],
    [
      b. $P(6,3)$
      #solve[$6! / (6 - 3)! = 120$]
    ],
    [
      c. $P(12,9)$
      #solve[$12! / (12 - 9)! = 79,833,600$]
    ],

    [
      d. $C(9,5)$
      #solve[$9! / (5! ⋅ (9 - 5)!) = 120$]
    ],
    [
      f. $C(12,7)$
      #solve[$12! / (7! ⋅ (12 - 7)!) = 792$]
    ],
  )

6. How many ways are there to select a first-prize winner, a second-prize winner, and a third-prize winner from 180 different people who have entered a contest? (6 pts)

  #note[$180 ⋅ 179 ⋅ 178  = #solvein[5,735,160]$]

7. How many bit strings of length 16 contain... (12 pts)

  #grid(
    columns: 2,
    align: center,
    gutter: 2em,
    [
      a. Exactly five 0s?
      #solve[$16! / (5! ⋅ (16 - 5)!) = 4,368$]
    ],
    [
      b. At most five 0s?
      #note[
        $
          &C(16,0) &= &16! / (0! ⋅ (16 - 0)!)& &= &1\
          + &C(16,1) &= &16! / (1! ⋅ (16 - 1)!)& &= &16\
          + &C(16,2) &= &16! / (2! ⋅ (16 - 2)!)& &= &120\
          + &C(16,3) &= &16! / (3! ⋅ (16 - 3)!)& &= &560\
          + &C(16,4) &= &16! / (4! ⋅ (16 - 4)!)& &= &1,820\
          + &C(16,5) &= &16! / (5! ⋅ (16 - 5)!)& &= &4,368\
        $
        #align(center)[#solve[$6,885$]]
      ]
    ],

    [
      c. At least five 0s?
      #note[
        $
          2^16 - 6,885 = #solvein[58,651]
        $
      ]
    ],
    [
      d. An equal number of 0s and 1s?
      #note[
        $
          C(16,8) = 16!/(8! ⋅ (16 - 8)!) = #solvein[12,870]
        $
      ]
    ],
  )
#align(center + bottom, note[#text(size: 3em)[⇊ SEE NEXT PAGE ⇊]])
#pagebreak()

8. How many permutations of letters ${"A", "B", "C", "D", "E", "F", "G", "H", "I", "J"}$ contain (no letters repeated)... (12 pts)

#grid(
  columns: 2,
  gutter: 1em,
  inset: (left: 2em, right: 2em),
  [
    a. The string AJ?
    #note[
      Our set is ${{"AJ"}, "B", "C", "D", "E", "F", "G", "H", "I"}$.

      Total of 9 permutations: $9! = #solvein[362,880]$
    ]
  ],
  [
    b. The string BIG?
    #note[
      Our set is ${{"BIG"}, "A", "C", "D", "E", "F", "H", "J"}$.

      Total of 8 permutations: $8! = #solvein[40,320]$
    ]
  ],

  [
    c. The strings AEG and DB?
    #note[
      Our set is ${{"AEG"}, {"DB"}, "C", "F", "H", "I", "J"}$.

      Total of 7 permutations: $7! = #solvein[5,040]$
    ]
  ],
  [
    d. The strings BG, FC, and AE?
    #note[
      Our set is ${{"AE"}, {"BG"}, {"FC"}, "D", "H", "I", "J"}$.

      Total of 7 permutations: $7! = #solvein[5,040]$
    ]
  ],

  [
    e. The strings FIG and GIF?
    #note[
      FIG = GIF

      Our set is ${"A", "B", "C", "D", "E", {"GIF"}, "H",  "J"}$.

      Total of 8 permutations: $8! = #solvein[40,320]$
    ]
  ],
  [
    f. The strings CD and DJ?
    #note[
      CD and DJ overlap, thus the string can be considered CDJ.

      Our set is ${"A", "B", {"CDJ"}, "E", "F", "G", "H", "I"}$.

      Total of 8 permutations: $8! = #solvein[40,320]$
    ]
  ],
)

9. Suppose that a department contains 9 dentists and 15 optometrists. How many ways are there to form a committee with 7 members if it must have more dentists than optometrists? (8 pts)
  #note[
    $
      "1. 4 dentists and 3 optometrists" -> C(9,4) ⋅ C(15,3) &= 126 ⋅ 455 &=& 57,330\
      "2. 5 dentists and 2 optometrists" -> C(9,5) ⋅ C(15,2) &= 126 ⋅ 105 &=& 13,230\
      "3. 6 dentists and 1 optometrists" -> C(9,6) ⋅ C(15,1) &= 84 ⋅ 15 &=& 1,260\
      "4. 7 dentists and 0 optometrists" -> C(9,7) ⋅ C(15,0) &= 36 ⋅ 1 &=& 36\
      \
      \
      57,330 + 13,230 + 1,260 + 63  = #solvein[71,883]
    $
  ]

#align(center + bottom, note[#text(size: 3em)[⇊ SEE NEXT PAGE ⇊]])
#pagebreak()

10. What is the coefficient of $x^17 ⋅ y^14$ in $(2x - 3y)^31$? (You do not need to calculate the final value. Just write down the formula of the coefficient) (10 pts)

  #note[
    $x = 2x, y = -3y, (X + Y)^31$
    $
      &= mat(31; 17) X^17 ⋅ Y^14\
      &= mat(31; 17) (2x)^17 ⋅ (-3y)^14\
      &= mat(31; 17) (2)^17 ⋅ (-3)^14 x^17 ⋅ y^13\
      &= #solve[$-mat(31; 17) 2^17 ⋅ 3^14$]
    $
  ]