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doh re mi fa so la ti do - AssociationVoting

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 \frac{1}{2} \left( \dbinom{6}{0} + \dbinom{6}{1} + \dbinom{6}{2} + \dbinom{6}{3} \right) = \frac{1}{2} (1 + 6 + 15 + 20) = \frac{42}{2} = \boxed{21}

📌 Pregunta: Three integers are randomly selected from 1 to 50 inclusive. ¿Cuál es la probabilidad de que su suma sea divisible por 3?

📌 Solución: Para encontrar la probabilidad, primero determinamos el número de maneras de elegir tres números cuya suma es divisible por 3. Usando aritmética modular, los posibles residuos al dividir entre 3 son 0, 1, 2. Las combinaciones favorables son:

📌 Los tres números tienen el mismo residuo: \(\dbinom{17}{3} + \dbinom{17}{3} + \dbinom{16}{3}\) (ya que hay 17 números conresiduo 0, 17 conresiduo 1, y 16 conresiduo 2).

📌 Un número de cada residuo: \(17 \times 17 \times 16\).

📌 Número total de maneras de elegir 3 números: \(\dbinom{50}{3}\).

📌 Probabilidad:

📌 \frac{3\dbinom{17}{3} + \dbinom{16}{3} + 17 \times 17 \times 16}{\dbinom{50}{3}} = \frac{3 \times 680 + 560 + 4568}{19600} = \frac{2040 + 560 + 4568}{19600} = \frac{6168}{19600} = \boxed{\dfrac{1542}{4900}}

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