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Home » Solve the equation \(3(x - 2) + 4 = 2x + 10\). - Appfinity Technologies

Solve the equation \(3(x - 2) + 4 = 2x + 10\). - Appfinity Technologies

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: A museum has 7 digital kiosks arranged in a circle for interactive displays. In how many distinct ways can 4 curator-led sessions (each hosted by a different kiosk) be assigned to 4 kiosks, if rotations of the arrangement are considered identical?

📌 Solution: Since the kiosks are arranged in a circle and rotations are considered the same, we fix one session to eliminate rotational symmetry. First, assign 4 out of 7 kiosks to the sessions: $inom{7}{4} \cdot 4!$ would count linear arrangements. But due to circular symmetry, we divide by 4 (the number of rotations).

📌 Alternatively, fix one session at a position to break rotational symmetry. Then choose 3 remaining kiosks from the remaining 6 and assign the other 3 sessions:

📌 Number of ways: $inom{6}{3} \cdot 3! = 20 \cdot 6 = 120$.

📌 Hence, the number of distinct circular arrangements is $oxed{120}$.

📌 Question: An ice cream truck offers 8 organic fruit sorbets and 5 blend styles. A customer chooses 2 different sorbets and 1 blend style. How many distinct combinations are possible if the order of sorbets does not matter?

📌 Solution: Choose 2 sorbets from 8: $inom{8}{2} = rac{8 \cdot 7}{2} = 28$.

📌 Choose 1 blend style from 5: $inom{5}{1} = 5$.

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Now, add these results:
+ 5 + 1 + 3 = 12
Next, find the remainder of 12 when divided by 6:
\equiv 0 \pmod{6}
Thus, the remainder is \(\boxed{0}\).
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We seek integers \( n \) such that \( n \equiv 3 \pmod{7} \). The sequ...
This is an arithmetic sequence with the first term \( a = 3 \) and com...
n_k = 3 + 7(k-1) = 7k - 4
We require \( n_k \leq 50 \):

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