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8. Simplificar \( \frac{9!\cdot 6!\cdot 3!}{8!\cdot 5!\cdot 2!} \)

Ask by Johnston Bowen. in Colombia
Mar 15,2025

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Tutor-Verified Answer

Answer

La expresión simplificada es 162.

Solution

\[ \textbf{Paso 1: Simplificar utilizando la propiedad de los factoriales} \] Observamos que \(9! = 9 \cdot 8!\), por lo que la expresión \[ \frac{9! \cdot 6! \cdot 3!}{8! \cdot 5! \cdot 2!} \] se puede escribir como: \[ \frac{9 \cdot 8! \cdot 6! \cdot 3!}{8! \cdot 5! \cdot 2!} \] Cancelamos \(8!\) en numerador y denominador: \[ \frac{9 \cdot 6! \cdot 3!}{5! \cdot 2!} \] \[ \textbf{Paso 2: Expresar \(6!\) en función de \(5!\)} \] Sabemos que \(6! = 6 \cdot 5!\), entonces: \[ \frac{9 \cdot 6 \cdot 5! \cdot 3!}{5! \cdot 2!} \] Cancelamos \(5!\): \[ \frac{9 \cdot 6 \cdot 3!}{2!} \] \[ \textbf{Paso 3: Sustituir los valores de \(3!\) y \(2!\)} \] Recordamos que: \[ 3! = 6 \quad \text{y} \quad 2! = 2 \] Por lo tanto: \[ \frac{9 \cdot 6 \cdot 6}{2} \] \[ \textbf{Paso 4: Realizar las operaciones} \] Multiplicamos los números: \[ 9 \cdot 6 \cdot 6 = 9 \cdot 36 = 324 \] Dividimos: \[ \frac{324}{2} = 162 \] La expresión simplificada es: \[ \boxed{162} \]

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Para simplificar \( \frac{9!\cdot 6!\cdot 3!}{8!\cdot 5!\cdot 2!} \), primero podemos reescribir los factoriales: \[ 9! = 9 \cdot 8!, \quad 6! = 6 \cdot 5!, \quad 3! = 3 \cdot 2! \] Sustituyendo estos en la expresión original, tenemos: \[ \frac{9 \cdot 8! \cdot 6 \cdot 5! \cdot 3 \cdot 2!}{8! \cdot 5! \cdot 2!} \] Ahora, observamos que \( 8! \), \( 5! \) y \( 2! \) se cancelan en el numerador y denominador: \[ = 9 \cdot 6 \cdot 3 \] Calculamos eso: \[ = 9 \cdot 6 = 54 \quad \text{y} \quad 54 \cdot 3 = 162 \] Por lo tanto, el resultado de la simplificación es: \[ \frac{9!\cdot 6!\cdot 3!}{8!\cdot 5!\cdot 2!} = 162 \]

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