• Calcolo Combinatorio E Probabilita -italian Edi... 🎉

    Every Saturday, Enzo offered a — a mystery pizza with random toppings chosen by a strange ritual. Customers would write their names on slips of paper, and Enzo would draw three names. Those three would each choose a topping from a list of ten: funghi, carciofi, salsiccia, peperoni, olive, cipolle, acciughe, rucola, gorgonzola, zucchine .

    Just then, the bell rang. Three new customers entered: a nun, a clown, and a beekeeper.

    In the narrow, lantern-lit streets of Perugia, old Enzo ran the most beloved pizzeria in Umbria. But Enzo had a secret: he was also a mathematician who had retired early from the University of Bologna.

    Total possible ordered selections (without replacement from 20): (20 \times 19 \times 18 = 6840). Calcolo combinatorio e probabilita -Italian Edi...

    Thus, overall probability that a pizza is made the customers are from three different towns: [ \frac{9}{10} \times \frac{25}{57} = \frac{225}{570} = \frac{45}{114} = \frac{15}{38} \approx 0.3947 ] The Revelation Chiara finished her wine. "Enzo, your pizza game is a lesson in combinatorics and probability."

    "Now that’s combinations without repetition for the selection, but with permutations for the picking order," Enzo explained.

    [ \frac{720}{1000} = 0.72 \quad (72%) ]

    "So most of the time," Marco laughed, "the pizza is a mix of three distinct flavors!" That night, a boy named Luca asked the most curious question: "What if you drew the names without replacement from a total of 20 customers, but then the three chosen still pick toppings with repetition? And also, before picking toppings, you shuffle a deck of 40 Scoppia cards (Italian regional cards: four suits, numbered 1 to 10). If the first card is a '1' of any suit, you cancel the pizza game. If not, you proceed. What’s the chance we actually make a pizza?"

    This is always possible once we reach this stage. So the probability that a pizza gets made is just the probability of not drawing a '1' first:

    "So," Chiara said, "a 1% chance. Rare, but possible." Every Saturday, Enzo offered a — a mystery

    "Enzo," she said, "what’s the probability that the three chosen customers all pick the same topping?"

    10 possible choices (all mushrooms, all onions, etc.) [ \frac{10}{1000} = \frac{1}{100} ]

    Choose 1 from town A: 5 ways, 1 from B: 5, 1 from C: 5, 1 from D: 5, but we need exactly 3 towns — so first choose which 3 towns out of 4: (\binom{4}{3} = 4) ways. For each set of 3 towns: choose 1 person from each: (5 \times 5 \times 5 = 125) combinations. Then arrange them in order: (3! = 6) ways. Total favorable ordered selections: [ 4 \times 125 \times 6 = 3000 ] Just then, the bell rang

    Enzo nodded. "It happened once. A trio of truffle enthusiasts. The pizza was… intense." A burly farmer named Marco asked, "What about the chance that all three toppings are different?"

Every Saturday, Enzo offered a — a mystery pizza with random toppings chosen by a strange ritual. Customers would write their names on slips of paper, and Enzo would draw three names. Those three would each choose a topping from a list of ten: funghi, carciofi, salsiccia, peperoni, olive, cipolle, acciughe, rucola, gorgonzola, zucchine .

Just then, the bell rang. Three new customers entered: a nun, a clown, and a beekeeper.

In the narrow, lantern-lit streets of Perugia, old Enzo ran the most beloved pizzeria in Umbria. But Enzo had a secret: he was also a mathematician who had retired early from the University of Bologna.

Total possible ordered selections (without replacement from 20): (20 \times 19 \times 18 = 6840).

Thus, overall probability that a pizza is made the customers are from three different towns: [ \frac{9}{10} \times \frac{25}{57} = \frac{225}{570} = \frac{45}{114} = \frac{15}{38} \approx 0.3947 ] The Revelation Chiara finished her wine. "Enzo, your pizza game is a lesson in combinatorics and probability."

"Now that’s combinations without repetition for the selection, but with permutations for the picking order," Enzo explained.

[ \frac{720}{1000} = 0.72 \quad (72%) ]

"So most of the time," Marco laughed, "the pizza is a mix of three distinct flavors!" That night, a boy named Luca asked the most curious question: "What if you drew the names without replacement from a total of 20 customers, but then the three chosen still pick toppings with repetition? And also, before picking toppings, you shuffle a deck of 40 Scoppia cards (Italian regional cards: four suits, numbered 1 to 10). If the first card is a '1' of any suit, you cancel the pizza game. If not, you proceed. What’s the chance we actually make a pizza?"

This is always possible once we reach this stage. So the probability that a pizza gets made is just the probability of not drawing a '1' first:

"So," Chiara said, "a 1% chance. Rare, but possible."

"Enzo," she said, "what’s the probability that the three chosen customers all pick the same topping?"

10 possible choices (all mushrooms, all onions, etc.) [ \frac{10}{1000} = \frac{1}{100} ]

Choose 1 from town A: 5 ways, 1 from B: 5, 1 from C: 5, 1 from D: 5, but we need exactly 3 towns — so first choose which 3 towns out of 4: (\binom{4}{3} = 4) ways. For each set of 3 towns: choose 1 person from each: (5 \times 5 \times 5 = 125) combinations. Then arrange them in order: (3! = 6) ways. Total favorable ordered selections: [ 4 \times 125 \times 6 = 3000 ]

Enzo nodded. "It happened once. A trio of truffle enthusiasts. The pizza was… intense." A burly farmer named Marco asked, "What about the chance that all three toppings are different?"