Combinatorics and Jason’s Deli (Part 2)

Jason’s Deli is one of my family’s favorite places for an inexpensive meal. Recently, I saw the following placard at our table advertising their salad bar:

The small print says “Math performed by actual rocket scientist”; let’s see how the rocket scientist actually did this calculation.

The advertisement says that there are 50+ possible ingredients; however, to actually get a single number of combinations, let’s say there are exactly 50 ingredients. Lettuce will serve as the base, and so the 5 ingredients that go on top of the lettuce will need to be chosen from the other 49 ingredients.

Also, order is not important for this problem… for example, it doesn’t matter if the tomatoes go on first or last if tomatoes are selected for the salad.

Therefore, the number of possible ingredients is

$\displaystyle {49 \choose 5}$ ,

or the number in the 5th column of the 49th row of Pascal’s triangle. Rather than actually finding the 49th row of Pascal’s triangle by direct addition, it’s simpler to use factorials:

$\displaystyle {49 \choose 5} = \displaystyle \frac{49!}{5! \times 44!} = \displaystyle \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44!}{5 \times 4 \times 3 \times 2 \times 1 \times 44!}$

$= \displaystyle \frac{49 \times 48 \times 47 \times 46 \times 45}{5 \times 4 \times 3 \times 2 \times 1}$

$= 49 \times 12 \times 47 \times 23 \times 3$

$= 1,906,884$ .

Under the assumption that there are exactly 50 ingredients, the rocket scientist actually got this right.

Combinatorics and Jason’s Deli (Part 1)

Jason’s Deli is one of my family’s favorite places for an inexpensive meal. Recently, I saw the following placard at our table advertising their salad bar:

I share this in the hopes that this might be reasonably engaging for students learning about different methods of counting.

Mathematics: It’s All Good

Source: http://maa-store.hostedbywebstore.com/MAA-ALL-GOOD-T-SHIRT/dp/B008VGK9NS

Predicate Logic and Popular Culture (Part 92): Annie Get Your Gun

Let $W(x,y)$ measure how well $x$ can do $y$ . Translate the logical statement

$\forall y (W(\hbox{you},y) < W(\hbox{I},y))$ .

This needs no further introduction:

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 91): The Scarlet Pimpernel

Let $H(x,t)$ be the proposition “ $x$ happens at time $t$ ,” let $V(x)$ be the proposition “ $x$ is a valley,” let $M(x)$ be the proposition “ $x$ is a mountain,” let $S(x,y,t)$ be the proposition “ $y$ must scale $x$ at time $t$ ,” let $W(x)$ be the proposition “ $x$ are perilous waters,” and let $S(x,y,t)$ be the proposition “ $y$ must sail $x$ at time $t$ .” Translate the logical statement

$\forall t \exists x (V(x) \land H(x,t)) \land \forall t \exists x(M(x) \land H(x,t) \land \exists y S(x,y,t))$

$\land \forall t \exists x(W(x) \land H(x,t) \land \exists y S(x,y,t))$ .

This matches the second half of the opening verse of the showstopper “Into The Fire” from the musical The Scarlet Pimpernel.

I also recommend Steve Amerson’s rendition:

Predicate Logic and Popular Culture (Part 90): The Beatles

Let $N(x)$ be the proposition “You need $latex x.” Translate the logical statement

$N(\hbox{love}) \land \forall x (x \ne \hbox{love} \Rightarrow \lnot N(x))$ .

This matches the title of one of the Beatles’ greatest hits.

Predicate Logic and Popular Culture (Part 89): Into the Woods

Let $S(x)$ be the proposition “ $x$ is on your side” and let $A(x)$ be the proposition “ $x$ is alone.” Translate the logical statement

$(\exists x S(x)) \land (\forall x \lnot A(x))$ .

Of course, this matches the last two lines of “No One Is Alone,” which was released as a movie in 2014.

I’m personally partial to Steve Amerson’s rendition of this song:

Predicate Logic and Popular Culture (Part 88): Bob Dylan

Let $H(x)$ be the proposition “You have $x$ ,” let $L(x)$ be the proposition “You have $x$ to lose,” let $p$ be the proposition “You’re invisible now,” and let $S(x)$ be the proposition “ $x$ is a secret to conceal.” Translate the logical statement

$((\forall x \lnot H(x)) \Rightarrow (\forall x \lnot L(x))) \land p \land \forall x (S(x) \Rightarrow \lnot H(x))$ .

This matches the last two lines of the closing verse of this classic from Bob Dylan.

Predicate Logic and Popular Culture (Part 87): Bruce Springsteen

Let $R(x)$ be the proposition “ $x$ needs a place to rest,” let $H(x)$ be the proposition “ $x$ wants to have a home,” and let $A(x)$ be the proposition “ $x$ wants to be alone.” Translate the logical statement

$\forall x (R(x) \land H(x) \land \lnot A(x))$ .

This matches three of the lines in the closing verse of one Bruce Springsteen’s great hits from the 1980s.

Predicate Logic and Popular Culture (Part 86): My Fair Lady

Let $T(t)$ be the proposition “ $t$ is at night” and let $D(t)$ be the proposition “I could have danced at time $t$ .” Translate the logical statement

$\forall t (T(t) \Rightarrow D(t))$ .