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Home » Now compute $ 8104 \mod 10 $: - Fine VPN

Now compute $ 8104 \mod 10 $: - Fine VPN

📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 Solution: Total letters: 26. Vowels: 5, consonants: 21. Total 3-letter arrangements with distinct letters: $26 imes 25 imes 24$. Unfavorable cases (no vowels): $21 imes 20 imes 19$. Favorable probability: $1 - rac{21 imes 20 imes 19}{26 imes 25 imes 24} = 1 - rac{7980}{15600} = 1 - rac{133}{260} = rac{127}{260}$. $oxed{\dfrac{127}{260}}$

📌 Question: If a historian selects 3 manuscripts at random from a collection of 15, where 5 are from the 18th century, what is the probability that all 3 selected are 18th-century manuscripts?

📌 Solution: The number of ways to choose 3 18th-century manuscripts is $inom{5}{3}$. Total ways to choose 3 manuscripts from 15 is $inom{15}{3}$. The probability is $rac{inom{5}{3}}{inom{15}{3}} = rac{10}{455} = rac{2}{91}$. $oxed{\dfrac{2}{91}}$Question: If the combined population of two marine mammal species in a region is 10 and the sum of their squared populations is 50, find the sum of their cubes.

📌 Solution: Let the populations be $ a $ and $ b $. Given $ a + b = 10 $ and $ a^2 + b^2 = 50 $. First, compute $ ab $ using $ (a + b)^2 = a^2 + 2ab + b^2 $. Substituting, $ 100 = 50 + 2ab $, so $ ab = 25 $. The sum of cubes is $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) = 1000 - 3 \cdot 25 \cdot 10 = 1000 - 750 = oxed{250} $.

📌 Question: A financial model predicts revenue from two products with $ x + y = 4.5 $ and $ x^2 + y^2 = 10.25 $. What is $ 5x + 5y $?

📌 Solution: Given $ x + y = 4.5 $, multiplying by 5 yields $ 5x + 5y = 5(x + y) = 5 \cdot 4.5 = oxed{22.5} $.

📌 Question: Rationalize the denominator of $ rac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} $.

📌 Solution: Multiply numerator and denominator by $ \sqrt{7} + \sqrt{3} $:

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El área de la superficie se calcula como 2(lw + lh + wh) = 2(8*5 + 8*3...
#### 158Pregunta: Si $x + y = 6$ y $x^2 + y^2 = 20$, encuentra $x^3 +...
Solución:
Primero, utilizamos la identidad para la suma de cubos:
x^3 + y^3 = (x + y)(x^2 - xy + y^2)
Ya conocemos $x + y = 6$ y $x^2 + y^2 = 20$. Para usar la identidad, n...
Sustituyendo los valores conocidos:
6^2 = 20 + 2xy \implies 36 = 20 + 2xy \implies 2xy = 16 \implies xy =...
Ahora sustituimos de nuevo en la identidad de la suma de cubos:
x^3 + y^3 = (x + y)(x^2 - xy + y^2) = 6(20 - 8) = 6 \times 12 = 72

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