GingerAle

Solutions:

a) John has the tiles A, B, E, E, P, T and Z on his tile rack. How many words of seven tiles can he make if he can also make nonsense words like BEEAPTZ ?

Answer: He can make (7!/2!) = 2520 words.  The two (2) “E” letters are indistinguishable and the redundancies are factored out by dividing by the factorial(s) of the duplicates.

b) Carlos has the tiles E, F, O, O, R and R on his tile rack. How many words of six tiles can he make if he can also make nonsense words?

Answer: (6!/2!/2!) = 180 words. This is the same algorithm as described above.

 c) How many (nonsense) words can Carlos make that begin with a vowel?

Answer: This would depend on the number of tiles, the number of vowels, and the number of duplicates Carlos has on his rack.  With the sample space of E, F, O, O, R and R, then starting with E,  5!/2!/2! = 30. Starting with O, 5!/2! =60. The sum of these two sets is the number of words he can make.  

d) In the bag there are ten more tiles. They are all different. John's tile rack is empty so he draws five more tiles from the bag. He puts them on the tile rack in the order he drew them. How many different (nonsense) words can John get on his tile rack?

Answer: With all distinguishable tiles, then nPr(10, 5) = 30240 different (nonsense) words.

GA

Solution:

\(\text {There are $\mathbf{(5-1)!}$ ways to arrange the boys in a circle. } | \small \text {(circular permutation) }\\ \text {There are five $\mathbf{(5)}$ positions between the boys. }\\ \text {There are $\mathbf{P(5,3)}$ ways to arrange the girls to occupy the positions between the boys.}\\ \mathbf{(4!) * P(5,3) = }\\ \mathbf{(24) * (60) = 1440} \\ \text{ }\\ \large \text {There are $\mathbf{1440}$ unique circular arrangements of 5 boys and 3 girls,} \\ \large \text {where each girl is separated from the other girls by at least one boy. }\\ \)

GA

Hi Melody,

I’ve read your post in detail and I understand your logic and methodology. It’s consistent with Rom’s view and solution process.

It still seems odd to me that the output sets, which are distinguishable by observation (because of the color), are factored out and discounted when the numeric element is redundant.  I know that sometimes certain elements that make a set unique are ignored because it’s irrelevant to the question. Similarly, some sets that are truly indistinguishable are treated as distinguishable because it affects the probabilities of their incidence.

For most counting problems, it seems straightforward for when to count or not to count redundancies.  For counting questions of this type, I suppose it’s a matter of practiced experience to understand the nuances for when to include or exclude redundancies.  Considering that both you and Rom have solved these problems for years and years, long before I was born, I accept the consensus that the general solution should factor out redundancies, (only counting them if an explicit condition in the problem requires it).

Thank you for your input on the solution to this question.

GA 

Solution:

\(\small \text{There are } \dbinom{16}{4} = 1820 \text{ sets of 4-cards. Four of these sets contain exactly one (1) card with a one.}\\ \text {Probability: } \dfrac{4}{1820} = \dfrac{1}{455}\\\)

GA

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I’m leaving this train wreck of a solution here as a sign post to slow down the trains of thought, before changing tracks. 

I initially solved this using (4^4) as the number of arrangements where the “1 cards” could occupy any row in the four columns of four rows (this is easy to see). Then dividing this by the total number of combinations,(4⁴) / (nCr(16,4)), gives (64/445) = 0.1406593406593407, the probability.  

For some reason, I thought this was too high of a probability.... 

So, what did I do to “fix” it? I’m not sure ... took the fourth root before dividing it (maybe –IDFK). After changing tracks, I didn’t realize the train actually jumped the tracks until the caboose ended up where the engine should be.  This is a simple, high school level probability problem that my cat could solve when buzzed on catnip, yet I turned it into a bloody train wreck. 

I am a much better troll than I am a mathematician.

Melody’s solution is defined and nuanced at every step. It’s even better than my cat could do.

GA

^{Guest wrote}: ^{I love the image of the sniper morphing into Nikita Khruschev and pounding his shoe... funny, and prophetic too.} 

I was totally LMAO when I wrote that. 

As long as we're devising alternatives, why not simply omit the sentence?  Don't have the sniper throw down the revolver at all. 

Not throwing down the gun would deprive the author of an optimal device for depicting the emotional state of the character’s transition from intently focused sniper to the pinnacle of maniacal rage when throwing the revolver, back to cogent reality after its discharge and near-miss of his head. His laughter, after the near-miss, reinforces the point.  The gun is the optimal literary device for this, despite the glaring error in the verisimilitude.

Of course, that deprives us of the opportunity to second guess the author for his error about the firearm discharging.  

It’s certainly possible the author was aware of this but chose to ignore it for the sake of the story. 

However, there are other obvious errors in verisimilitude that I didn’t mention. However, they might only be obvious to those familiar with military history and weaponry, ballistics, and basic human physiology.   

There was a small hole where the bullet had entered. On the other side there was no hole. The bullet had lodged in the bone. It must have fractured it.

It is very improbable for the bullet to have lodged in the forearm bone (or any bone) of the militia man. It’s logical to assume the rifles used by both snipers were either the Pattern 1914, standard issue for British military troops, or the Ross Mk III (preferred by snipers).   Both these weapons fired caliper .303 copper-nickel jacketed lead- ball ammunition.  When fired, the bullet has 3500 Joules of kinetic energy exiting the muzzle. At fifty yards, there is still enough energy to travel through the forearm bones of atleast six large-boned men.   Snipers were known to cut or score the bullet tips to promote its expansion on impact, increasing the likelihood of a kill or serious wounding. If the bullet expanded on impact, it still wouldn’t stop in the bone, it would continue on, taking a large portion of the arm with it –disgusting, but an accurate approximation.    

Other inconsistencies with the revolver:

“He was almost deafened with the report and his arm shook with the recoil.”

This suggests a large caliper sidearm. There are a few choices for this weapon:

The .45 caliper M1917 Revolver

The .455 caliper Webley Revolver Mk VI

Both of these revolvers have a barrel length of more than half overall length of ~11 inches.

 “He picked up his revolver and put it in his pocket. Then he crawled down through the skylight to the house underneath.”

These revolvers would not conveniently fit in a pocket. It would fall out after three steps. This suggests a smaller sidearm. In any case, a trained military combatant would never do this. If his holster wasn’t available, he’d stick it in his waistband, being intuitively careful not to turn himself into a eunuch. 

-----

He was now standing before a row of chimney pots, looking across, with his head clearly silhouetted against the western sky.

A trained military combatant –especially a sniper –would never expose his position in this manner.  This is just plain stupid...

It’s easy to see the author was unfamiliar with military tactics and weapons.  He was however very familiar with pens, typewriters, and biblically based moral precepts because of his education and training for the priesthood. A notable irony, here: the author was probably already an atheist when he wrote this story, having abandoned his divinity studies and then his faith, a few years earlier.     

GA

^{Oh, yes, it is.  It's very much different.  The golfer misses an easy putt, and the carpenter hits his thumb.  Both failures, frustrating instances that should not have happened.  But the sniper had successfully defeated his enemy, his deadly enemy, and by making a difficult shot with a handgun,...}

For sure, my analogous examples lack the requisite success elements for this comparison. However, the main question is, did the author use the protagonist’s aggression towards the revolver to blame the gun for his actions? Or, was it a device use to convey rage? 

There wasn’t much else to use for depicting rage: his rifle was gone, so was his hat. Perhaps he could use his boot. Let’s see how that works.

The regret and crescendo of pain induced shock took its toll on the warrior; in a rage, he removed his boot and pounded it on the roof with many curses and oaths. After he calmed himself, he decided to change his name to Nikita Khrushchev, and establish a communist party in Éire, so there!

Humm... It doesn’t really have much punch, but it does make for good satire.  (The author of this story, Liam O'Flaherty, was a founding member of the Communist Party of Ireland.) What else could we try? 

Let’s try foot-stomping.

The regret and crescendo of pain induced shock took its toll on the warrior; in a rage, he stomped his right foot so hard into the roof that he fell in up to his waist. Then with both hands, he took hold of his left foot and ripped himself up the middle in two.

Humm... This one is funny too, and Rumpelstiltskin gets a reprieve.   However, our protagonist is now dead, which means he cannot brave machinegun fire and conveniently discover that he killed his brother.   This kinda wrecks the end of the story. 

So, only the smoking gun remains.  The author has to risk an unintended double-action for the revolver: the readers interpreting it as a scapegoat, instead of the whipping-boy that it is.

Oh well, no one ever said it’s easy being a writer.

GA

i had a meltdown cuz im asian, and I worry about my future.

I’m sure your family’s culture has a great influence on your psychology. Being 13-years in age has a great affect too.

You might consider learning meditation techniques. Known as biofeedback in psychophysiology, these techniques have their origins mostly from ancient Asian philosophies. They really do work.  I have watched practitioners of these meditation techniques drop their heart rate by 15% in less than 5 minutes; their corresponding body temperature drops too, though it takes longer. Anyone who can control their sympathetic and parasympathetic nervous system with meditation can control their emotional states. I can personally attest that this is true, and I am a long way from being a master of these practices.    

Don't worry the teacher understands I didn't get in trouble.

I’m not surprised. Most teachers are not dumb –though there are some who more than qualify for the label of BatShit-Stupid. 

The teacher also told a spine-chilling story on how a kid last year legit Ctrl - c, ctrl v their friend's analysis paper from another year. A cretinous mistake it was, and she recieved a 0 on it.

(Cretinous. What a great word! I’ve noticed that archaic word usage has generally increased in the past few years.)

She ended up getting rejected from honor classes, which subsequently "ruined" her life.

Yes.  Cheating and honor are two principles of behavior that tend toward annihilating each other, and such releases of energy can certainly ruin a life.

********************

In prior years, I served three terms on the Scarlet Honor Council, my University’s Hearing Board for Academic. Though the honor code covered many topics, the cases involving plagiarism and other forms of cheating usually topped the list.  There were two instances of cheating that stood out as exceptional and odd in the manner of how the circumstances of these events unfolded into a bazaar reality for the accused cheaters. So much so, I recorded the details in my personal journal and log.  Condensations of these events from these excerpts are below. Most of the details are kept vague because of the confidential nature. 

In the first of these cases, a professor accused a student of overt plagiarism: copying 72% of his writing nearly word for word from two separate texts. This type of plagiarism is extremely rare at the university level, especially from a third-year student, and this student’s age was many years above the average for an undergrad.  What made this truly unusual was the accused student committed this plagiarism during an in class writing assignment.

Initially, the professor assumed the student had substituted a prewritten essay book, but then later said the student had memorized the passages of text.  Substitution of essay books is nothing new; most claim they just needed more than the allotted time to write in the class. However, most students who do this do not overtly plagiarize when writing their essays.

The professor discovered the plagiarism while reviewing the assignment for grading, and instantly recognizing the text from two separate books related in subject, with first editions published about five decades prior; both texts were available in the university library, and were listed for recommended reading for two of the professor’s classes, but not for the class the accused student was enrolled. The professor had his assistant type the essay for conversion to a digital format and scanned for plagiarism. The scan returned a 72% score for duplication. The professor summarily dropped the student from his class. This defaults to a withdrew failing.

The student acknowledged the plagiarism, saying there were extenuating circumstances. 

Would you like to guess what the extenuating circumstances were?

In the other case, five students of a study group made nearly perfect scores on an extremely difficult astrophysics test. These five were accused of stealing a copy of the test. Four of the five students were the top students in the professor’s class. One admitted to the theft, saying the others had no foreknowledge of the theft.  But they did know on test day, and they all benefited from it.   How would you judge this case?

GA

The author of this story, published in 1922, depicts the verisimilitude of the environment and the thought process of the protagonist in narrative form, from a third-person, limited omnipresent view point. The target audience was the naïve, unsophisticated, population of Ireland, whose average education ranged from the sixth to eight grades. 

In this era, prior to broadcast radio, world events were typically conveyed via a single newspaper and the silent films of Fox News (which became Fox Movietone News in the “talky” era) and repeated rumor in social circles, which were typically a church setting for most of the population, and, mainly for adult males, the pubs were another venue.   Complex concepts, such as the nuances of war and its effects on warriors and the population were otherwise conveyed to the population in various literary forms.  

Literature often describes civil wars as “brother against brother,” describing them figuratively and depicting them literally in many notable works.

The Sniper follows this form. This short story has an intense, juvenile cadence to it. Simple sentences convey uncomplicated, but highly focused thoughts of a militia warrior fighting a guerilla-style battle; and, after his injury, demonstrating behaviors consistent with battle fatigue and shell-shock that is already morphing into what, years later, would become known as PTSD. 

The author, who had previously studied for the priesthood, conveyed the humanity still present, and now coming to the surface, in this wounded warrior. Only a true psychopath or sociopath is immune to the psychological effects of killing another human, no matter how justified the action: “The lust of battle died in him. He became bitten by remorse. ...Weakened by his wound and the long summer day of fasting...”   The psychological effects of his injury and pain bring to mind the unspoken biblical aphorism of “all they that take the sword shall perish with the sword.”

The regret and crescendo of pain induced shock took its toll on the warrior; in a rage, he retaliated against his revolver, an iconic metaphor of his last homicide and a contributor, by way of recoil, to his pain.   The revolver retaliated:  “The revolver went off with a concussion and the bullet whizzed past the sniper's head.” bring the warrior back to his present reality.  His remorse would return when he found that his target was his brother.  This wouldn’t stop the war, or the protagonist’s participation –except for a few minutes while he mourns.   

Other comments:

The reality is that revolver would not have fired when he threw it down. The hammer was resting on an empty shell, previously fired at his target.

There are only two ways for the weapon to fire in this circumstance:

The militia man cocked it, which, along with being incredibly stupid, means he would not have been surprised by its discharge.

The other requires a sustained force on the trigger, equal to 10 to 15 pounds (44 to 67 Newtons) (typical for the firing of double action revolvers of this era, and still common today). This is extremely unlikely to happen.

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I've seen it in a dozen movies.  It's a typical and clichéd portrayed reaction to having used a firearm.  Shooting someone was justified and necessary, but the user feels remorse – or maybe guilt... whatever – and blames the gun for what he himself had done.  What a crock. 

Such clichéd depictions of remorse and guilt may common in movies and literature, and blaming “guns” is certainly common in the news media for gun violence.  However, in the case of this short story, there is no indication of this. The throwing of the gun is a depiction of frustrated anger and pain, not blame. It’s not much different than a depiction of a golfer throwing is club because he missed and easy putt or a carpenter throwing his hammer because he improperly framed a wall or hit his finger. 

GA

There is a way to graph the stars on a linear scale, but you’ll still not be able to identify the planets specifically.

Create a polar type graph with concentric circles, using an appropriate scale.

To use the distances listed in meters, start by placing a small circle at the center of the graphing paper label it “Earth’s sun” or whatever.

The sun with the greatest distance is Wolf 359 at 7.38E16 meters, round this up to 7.50E16 meters to allow for a little margin at the graph’s edge.

There are 12 lines above the sun and 12 lines below.  Divide 7.50E16 by 12.  

(7.50E16/12 = 6.25E15).

 6.25E15 is the number of meters the space between each line represents.

Divide each of the suns distances by 6.25E15 to find the number of grid lines from Earth’s sun.

Alpha Centaury: -- 4.16E16 / 6.25E15 = 6.66 grid lines

Barnard’s Star: --- 5.68E16 / 6.25E15 = 9.09 grid lines

Wolf 359 --------- 7.38E16 / 6.25E15 = 11.81 grid lines

Using a compass, with the sun as the origin, set the width to the number of grid lines, and then draw a circle around the sun. Do this for each of the stars.  These circles represent the distance of each star from the sum. 

Place a dot on each circle and label it accordingly. The stars are not in a straight line. Because the graph is polar, it’s easy to indicate the approximate direction of each star from Earth’s sun.  If you want to do that, here are the locations: Barnard’s Star is at the 30 degree mark, Wolf 359 is at 210 degrees, and Alpha Centaury is 310 degrees.

This information should be sufficient for you to create a linear graph of the nearest stars to Earth’s solar system.

GA