#set document(title: "7.1 The Multiplication Rule for Counting", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 7.1#h(0.6em)The Multiplication Rule for Counting #figure(figph[Close up of a hand lifting two playing cards at a poker table.], alt: "Close up of a hand lifting two playing cards at a poker table.", caption: [The Multiplication Rule for Counting allows us to compute more complicated probabilities, like drawing two aces from a deck.]) === Learning Objectives After completing this section, you should be able to: + Apply the Multiplication Rule for Counting to solve problems. One of the first bits of mathematical knowledge children learn is how to count objects by pointing to them in turn and saying: “one, two, three, …” That’s a useful skill, but when the number of things that we need to count grows large, that method becomes onerous (or, for #emph[very] large numbers, impossible for humans to accomplish in a typical human lifespan). So, mathematicians have developed short cuts to counting big numbers. These techniques fall under the mathematical discipline of #strong[combinatorics], which is devoted to counting. === Multiplication as a Combinatorial Short Cut One of the first combinatorial short cuts to counting students learn in school has to do with areas of rectangles. If we have a set of objects to be counted that can be physically arranged into a rectangular shape, then we can use multiplication to do the counting for us. Consider this set of objects : #figure(figph[A group of beach balls arranged in 1 long row.], alt: "A group of beach balls arranged in 1 long row.", caption: none) Certainly we can count them by pointing and running through the numbers, but it’s more efficient to group them . #figure(figph[A group of beach balls arranged in 4 rows of 6 balls.], alt: "A group of beach balls arranged in 4 rows of 6 balls.", caption: none) If we group the balls by 4s, we see that we have 6 groups (or, we can see this arrangement as 4 groups of 6 balls). Since multiplication is repeated addition (i.e., #math.equation(block: false, alt: "6 times 4 equals 4 plus 4 plus 4 plus 4 plus 4 plus 4")[$6 × 4 = 4 + 4 + 4 + 4 + 4 + 4$]), we can use this grouping to quickly see that there are 24 balls. Let’s generalize this idea a little bit. Let’s say that we’re visiting a bakery that offers customized cupcakes. For the cake, we have three choices: vanilla, chocolate, and strawberry. Each cupcake can be topped with one of four types of frosting: vanilla, chocolate, lemon, and strawberry. How many different cupcake combinations are possible? We can think of laying out all the possibilities in a grid, with cake choices defining the rows and frosting choices defining the columns (Figure 7.5). #figure( stdfig([#table( columns: (auto,) * 5, align: center+horizon, inset: 5pt, stroke: 0.5pt + black, table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[]), table.cell(fill: rgb("#f0f0f0"), align: center+horizon, colspan: 4, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[frostings]), table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[cakes]), table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[vanilla]), table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[chocolate]), table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[lemon]), table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[strawberry]), table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[vanilla]), [vanilla cake with vanilla frosting], [vanilla cake with chocolate frosting], [vanilla cake with lemon frosting], [vanilla cake with strawberry frosting], table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[chocolate]), [chocolate cake with vanilla frosting], [chocolate cake with chocolate frosting], [chocolate cake with lemon frosting], [chocolate cake with strawberry frosting], table.cell(fill: rgb("#f0f0f0"), align: center+horizon, text(font: sys.inputs.at("title-font", default: "STIX Two Text"), weight: "bold", size: 9pt)[strawberry]), [strawberry cake with vanilla frosting], [strawberry cake with chocolate frosting], [strawberry cake with lemon frosting], [strawberry cake with strawberry frosting], )]), alt: "A rectangular grid with 3 rows and 4 columns. The row headers representing the cakes show vanilla, chocolate, and strawberry. The column headers representing the frostings show vanilla, chocolate, lemon, and strawberry. Data from the grid are as follows. Row 1: vanilla cake with vanilla frosting, vanilla cake with chocolate frosting, vanilla cake with lemon frosting, and vanilla cake with strawberry frosting. Row 2: chocolate cake with vanilla frosting, chocolate cake with chocolate frosting, chocolate cake with lemon frosting, and chocolate cake with strawberry frosting. Row 3: strawberry cake with vanilla frosting, strawberry cake with chocolate frosting, strawberry cake with lemon frosting, and strawberry cake with strawberry frosting.", caption: none, ) Since there are 3 rows (cakes) and 4 columns (frostings), we have #math.equation(block: false, alt: "3 times 4 equals 12")[$3 × 4 = 12$] possible combinations. This is the reasoning behind the #strong[Multiplication Rule for Counting], which is also known as the Fundamental Counting Principle. This rule says that if there are #math.equation(block: false, alt: "n")[$n$] ways to accomplish one task and #math.equation(block: false, alt: "m")[$m$] ways to accomplish a second task, then there are #math.equation(block: false, alt: "n times m")[$n × m$] ways to accomplish both tasks. We can tack on additional tasks by multiplying the number of ways to accomplish those tasks to our previous product. #examplebox("Example 1")[Using the Multiplication Rule for Counting][ Every card in a standard deck of cards has two identifying characteristics: a suit (clubs, diamonds, hearts, or spades; these are indicated by these symbols, respectively: #math.equation(block: false, alt: "♣")[$♣$], #math.equation(block: false, alt: "♢")[$♢$], #math.equation(block: false, alt: "♡")[$♡$], #math.equation(block: false, alt: "♠")[$♠$]) and a rank (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king; the letters A, J, Q, and K are used to represent the words). Each possible pair of suit and rank appears exactly once in the deck. How many cards are in the standard deck? #solutionbox[ Since there are 4 suits and 13 ranks, the number of cards must be #math.equation(block: false, alt: "4 times 13 equals 52")[$4 × 13 = 52$] . #figure(figph[A deck of playing cards sorted by suit into 4 groups. Each group is arranged from largest to smallest value.], alt: "A deck of playing cards sorted by suit into 4 groups. Each group is arranged from largest to smallest value.", caption: [Standard Deck of Cards, Sorted by Rank and Suit]) ] ] #examplebox("Example 2")[Using the Multiplication Rule for Counting for 4 Groups][ The University Combinatorics Club has 31 members: 8 seniors, 7 juniors, 5 sophomores, and 11 first-years. How many possible 4-person committees can be formed by selecting 1 member from each class? #solutionbox[ Since we have 8 choices for the senior, 7 choices for the junior, 5 for the sophomore, and 11 for the first-year, there are #math.equation(block: false, alt: "8 times 7 times 5 times 11 equals 3,080")[$8 × 7 × 5 × 11 = 3,080$] different ways to fill out the committee. ] ] #examplebox("Example 3")[Using the Multiplication Rule for Counting for More Groups][ The standard license plates for vehicles in a certain state consist of 6 characters: 3 letters followed by 3 digits. There are 26 letters in the alphabet and 10 digits (0 through 9) to choose from. How many license plates can be made using this format? #solutionbox[ Since there are 26 different letters and 10 different digits, the total number of possible license plates is #math.equation(block: false, alt: "26 times 26 times 26 times 10 times 10 times 10 equals 17 , 576 , 000")[$26 × 26 × 26 × 10 × 10 × 10 = 17 , 576 , 000$]. ] ] === Key Terms - combinatorics - Multiplication Rule for Counting (Fundamental Counting Principle) === Key Concepts - The Multiplication Rule for Counting is used to count large sets.