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Home » Now, write in terms of \(c\). Use the identity: - Dachbleche24

Now, write in terms of \(c\). Use the identity: - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Pregunta: ¿De cuántas maneras se pueden distribuir 6 bolas distinguibles en 2 cajas indistinguibles?

📌 Solución: Cuando las cajas son indistinguibles, cada distribución se cuenta una vez, independientemente del orden de las cajas. El número de maneras es el número de particiones de 6 bolas en 2 grupos, donde el orden no importa. Esto se da por:

📌 \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: \(\dbinom{50}{3}\).

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