Archive for the ‘Combinations’ Category

The 1001 Arabian Nights

Thursday, November 23rd, 2017

The new harem of Caliph Mustafa ibn Mafeesh ben Rasheed – of which he was very proud – consisted of 14 particularly well-chosen prime virgins.

Having heard of the “Tales of 1001 Nights” and being a man who strongly believed in the cultural education of his people, Caliph Mustafa ibn Mafeesh ben Rasheed decided to have a different group of wives from his new harem entertained and edified each night with a story from this classical work.

Scheherazade, one of his older wives, was an enchanting story teller, so Caliph Mustafa ibn Mafeesh ben Rasheed decided she should be present each night to read a different story to the group selected from the new harem.

Caliph Mustafa wanted a different group of new wives to appear for 1001 nights so all the tales could be told for their cultural edification.

Problem was, he didn’t know what the size of the group should be so it would be different for each of the 1001 nights – not counting himself and Scheherazade. Caliph Mustafa ibn Mafeesh ben Rasheed wanted the group to be as large as possible to maximize the cultural education effort.

Can you help Caliph Mustafa ibn Mafeesh ben Rasheed work out the maximum size of the group so that for 1001 nights the composition of the group of new wives would be different?

The Christmas Gifts

Thursday, August 31st, 2017

Fred and Mary were trying to work out how to give five gifts to their three boys for Christmas.

The kids, Henry, Mark and Joe, had told them they wanted to receive gifts more or less at random this year to see how the gifts would be distributed.

“We have five gifts to give to three children. How are we going to do this, Mary? There are so many possibilities it makes my brain spin,” said Fred in exasperation.

“Well, Fred, we have to make sure they all get at least one gift, so that makes it easier than the way you are thinking of, doesn’t it,” Mary said reassuringly.

“You’re right, Mary. The number of ways of giving the gifts so that our kids might not receive even one gift is quite large,” said Fred.

“Do it my way and the matter becomes more simple,” said Mary.

So Mary and Fred distributed the gifts as she suggested.

If they distributed the gifts Fred’s way what would be the probability that one or more of the kids received no gift?

The Four Aces

Sunday, April 30th, 2017

Fred was playing poker with his three friends Henry, John and Mack and had just dealt a hand for each.

Fred saw that he had an Ace and wondered what the probability was that each of the other players also had an Ace.

Later in the evening after the game, Fred decided to write a computer program that would deal ten thousand hands of poker to check how often four players each would be dealt an Ace.

Then Fred wondered what would be the similar case for bridge hands dealt, for which purpose he changed a few input parameters for his computer program and received a surprise answer.

On the average, how many of the ten thousand poker hand deals do you think resulted in an Ace for each player?

On the average, how many bridge hand deals did Fred find gave an Ace to each player?

The Seating Arrangement

Friday, March 31st, 2017

A dating club arranges meetings at concerts for its members. The club will buy a certain number of random adjacent concert seats for some male and female members. If a man sits next to a woman, the club considers this a date opportunity.

For the coming concert, the club bought 13 adjacent seats for 7 men and 6 women. Some lucky ladies could give a card with their phone number to the man or men sitting next to them.

As an average estimate, about how many cards with phone numbers do you think were given by the ladies at the concert?

The Blind Mice

Thursday, February 23rd, 2017

“Hi Jack, how are things.” Jill sat down at the café table where Jack was enjoying a cup of coffee and eating a croissant as an afternoon snack to relax after university classes.

“Fine, Jill, but feeling a bit weird,” replied Jack, munching on his croissant.

“Tell me all about it, Jack, I’m listening,” said Jill. She ordered coffee and a donut from the hovering waitress.

“Well, I’m doing an internship for my course on statistics with a Dr. Schnitzelbrenner who is carrying out medical research using blind mice,” said Jack, sipping his coffee.

“And what does Dr. Schnitzelbrenner do with the blind mice,” asked Jill, adjusting the dishes on the table to accommodate the coffee and donut being carelessly dumped by the waitress.

“Well, Jill, he has a large black cat named Schreck – that seems to get fatter by the day – which he uses to frighten four blind mice into running out of their cages in the direction of five entrances, one containing a large chunk of their favorite aromatic cheese,” explained Jack.

“What in the world is Dr. Schnitzelbrenner doing that for,” asked an incredulous Jill. She moved closer to the table and crossed her legs, elbow on a knee holding up her head and gazing attentively at Jack.

“Dr. Schnitzelbrenner says he is testing the orientative capacity of terrorized blind mice compared with blind mice who are not in a state of terror, using a quadruple-blind testing method – all for the purpose of developing a drug for Big Pharma against disorientation,” said Jack nonchalantly as he was ingesting the last piece of croissant.

“How is this a quadruple-blind test,” inquired Jill.

“Well, Dr. Schnitzelbrenner is using four blind mice,” explained Jack.

“Oh, I get it, that’s why it’s a quadruple blind test,” laughed Jill.

“And how will Dr. Schnitzelbrenner determine whether his medicine works or not,” asked Jill.

“Well, if all the mice go into different entrances in any one trial, a light switches on and a buzzer sounds. It is my job as a statistician to record these trials, with or without signals going off,” explained Jack.

“According to Dr. Schnitzelbrenner, if the terrorized mice all head for the entrance with cheese, then they are not to be considered disoriented and the drug works. If they all go to different entrances, then they are randomly well disoriented,” continued Jack.

“Hmm, tell me more,” said Jill.

“If not too many mice die from the active substance in the drug, Dr. Schnitzelbrenner hopes to win the Nobel Prize for this new medicine, which he says will aid senior citizens who suffer from disorienting illnesses,” added Jack.

“Very quaint experiment, I must say, but senior citizens will not normally be in a state of terror,” observed Jill.

“According to Dr. Schnitzelbrenner, senior citizens actually do suffer from a suppressed subconscious fear-of-death-inspired terror, and that is really why they easily become disoriented,” explained Jack.

“I see. How long will you be working with Dr. Schnitzelbrenner, Jack?”

“Just another week or so, after I calculate the probability of randomly well-disoriented blind mice each winding up in separate entrances, which I find a bit difficult,” confessed Jack.

“How about a walk in the park, Jack, to get your mind off Dr. Schnitzelbrenner’s terrorizing research, and I’ll give you some hot tips,” she suggested, emptying her cup of coffee.

“Capital idea, Jill.”

They paid and left the café.

Can you work out the probability of a group of four blind mice all going into a separate entrance in any single trial with five entrances?

The First Ace

Tuesday, September 20th, 2016

Pierre, the First Mate, was sitting in the mess room of the Nautilus flipping cards from a Tarot deck. He would periodically stop and write some numbers in a small notebook near the cards.

“What are you doing, First Mate?” said Captain Nemo, who had stopped to look as he was walking by on his way to the bridge.

“Sir, I am checking to see how many cards on the average I need to draw from a shuffled Tarot pack before I get an Ace,” said Pierre with a bright smile as he drew The Hanged Man.

“Why don’t you just calculate it? It’s much faster,” grinned Captain Nemo.

“I thought about doing that, but there are so many combinations on how to get an Ace,” moaned Pierre scratching his head as he pulled The Fool.

“Why don’t you think about how not to get an Ace instead,” suggested Captain Nemo, continuing on his way to the bridge.

“Hmm, that’s not a bad idea, Sir,” the First Mate replied and started to write some symbols in his notebook.


What do you figure is the average number of Tarot cards that need to be drawn from a well-shuffled Tarot pack before an Ace appears?

Ice Cream Flavors

Sunday, August 14th, 2016

John was out with his three children, when Albert, the youngest, spotted “Guiseppe’s Italian Ice Cream Parlor.”

“Daddy, I want an ice cream,” whooped Albert, pulling his dad towards the ice cream shop with great enthusiasm and force.

“Yes, so do we,” shrieked Jenny and Mary, helping Albert drag their dad along.

John shrugged patiently. There was nothing for it but to follow and go see what Guiseppe had to offer.

“I can offer you seven flavors: chocolate, vanilla, strawberry, pistachio, stratiachiella, lemon and banana. Delicious home-made Italian ice cream. You will love it,” promoted Guiseppe cheerfully.

“If you choose a different combination each time you come, I give you one ice cream cone for free,” offered Guiseppe.

“Dad, I just figured it out. We can come here lots of times to get different combinations of flavors,” shouted Albert enthusiastically.

John groaned, a budding mathematician.


How many different combinations of flavors, one per ice cream cone, were possible for the four of them, even if one or more of them chose the same flavor?

And how many combinations if Mary always wanted the same flavor as Jenny, so long as nobody else wanted the same flavor?

The Round Table

Friday, July 8th, 2016

Jack and Jill were sitting in Banerjee’s Hindu Restaurant at a round table with seven persons, enjoying a delicious chicken tikki masala with spicy papadums and a fine wine, when the owner, a Mr. Banerjee, approached them with a wide smile.

“I am so happy to see such a handsome couple like you eating in my restaurant. To show my appreciation, I will set you a small task. If you can tell me how many different ways there are to arrange the seven people around this table, you of course in the same position, the meals at this table will be on the house. I will give you some time to consult. Please write down your answers on your bills. If correct, there will be no charge.”

What is the number needed for obtaining a free meal?

What would the number be if Jack and Jill were not required to sit together.

The Castle Moat

Friday, June 3rd, 2016

Sir Lancelot with his squire Jerkin wants to rescue the lovely princess Jennivere, kept prisoner by the Black Knight in his impregnable castle.

His obstacle is a rectangular moat full of poisonous dragon fish.

Looking about for useful resources, Jerkin finds two planks each of which is exactly as long at the moat is wide.

How will they get across without falling into the moat?

The Three Hats

Tuesday, May 24th, 2016

A billionaire enters a reality show saloon with five bellboy hats, three red and two white.

“I need three volunteers to play a game for a million dollars,” he said. “Any takers?”

Three men put down their beer mugs immediately and volunteered: Jack, Tobias and Charlie.

“Very well. Set up three chairs in a triangle so each of you can see the other two,” commanded the billionaire.

This was speedily arranged.

“Close your eyes while I place one of these five hats on each of your heads. I’ll tell you when to open them. The hats are so small you won’t be able to see the color of your own hat,” he explained cheerily.

He put the hats on their heads and threw the remaining two out the window.

“I’ll give a million dollars to the man who can say which color hat he has on his head,” he said.

They opened their eyes and looked eagerly at their respective companions.

“What do you say, Jack?” asked the billionaire.

“Ummm, I don’t know,” Jack said with a sad face.

“What about you, Tobias?” asked the billionaire.

Tobias looked studiously at his two companions. “Well, I can’t say either,” he grumbled.

“But I know,” exclaimed Charlie jubilantly, “and I am blind.”

And, sure enough, he did know and received the million dollars with a big smile.

What was the color of Charlie’s hat, and how did he know?