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Simplificar las siguientes expresiones:

A. [tex] \frac{12!}{10!} [/tex]

B. [tex] \frac{7!}{(7-2)!} [/tex]

C. [tex] \frac{8!}{(8-3)!3!} [/tex]


Sagot :

Claro, vamos a simplificar cada expresión paso a paso.

### Expresión A: \(\frac{12!}{10!}\)

Primero, recordemos la definición de factorial. \(12!\) significa \(12 \times 11 \times 10 \times 9 \times 8 \times \cdots \times 1\) y \(10!\) significa \(10 \times 9 \times 8 \times \cdots \times 1\).

[tex]\[ \frac{12!}{10!} = \frac{12 \times 11 \times 10!}{10!} \][/tex]

Aquí, los \(10!\) del numerador y del denominador se cancelan, dejando:

[tex]\[ 12 \times 11 = 132 \][/tex]

### Expresión C: \(\frac{7!}{(7-2)!}\)

\(7!\) es \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) y \((7-2)!\) es \(5!\) que es \(5 \times 4 \times 3 \times 2 \times 1\).

[tex]\[ \frac{7!}{(7-2)!} = \frac{7 \times 6 \times 5!}{5!} \][/tex]

Aquí, los \(5!\) del numerador y del denominador se cancelan, dejando:

[tex]\[ 7 \times 6 = 42 \][/tex]

### Expresión D: \(\frac{8!}{(8-3)!3!}\)

\(8!\) es \(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\), \((8-3)!\) es \(5!\) que es \(5 \times 4 \times 3 \times 2 \times 1\), y \(3!\) es \(3 \times 2 \times 1\).

Primero, simplifiquemos el numerador:

[tex]\[ 8! = 8 \times 7 \times 6 \times 5! \][/tex]

El denominador es:

[tex]\[ (8-3)! \times 3! = 5! \times 3! = 5! \times (3 \times 2 \times 1) \][/tex]

Entonces, la expresión se convierte en:

[tex]\[ \frac{8 \times 7 \times 6 \times 5!}{5! \times 3!} \][/tex]

Aquí, los \(5!\) del numerador y del denominador se cancelan, dejando:

[tex]\[ \frac{8 \times 7 \times 6}{3!} \][/tex]

Ahora, calculemos \(3!\):

[tex]\[ 3! = 3 \times 2 \times 1 = 6 \][/tex]

Entonces, la expresión se convierte en:

[tex]\[ \frac{8 \times 7 \times 6}{6} \][/tex]

Los \(6\) del numerador y del denominador se cancelan, dejando:

[tex]\[ 8 \times 7 = 56 \][/tex]

### Resultados Finales

[tex]\[ \begin{align*} A. & \quad \frac{12!}{10!} = 132 \\ C. & \quad \frac{7!}{(7-2)!} = 42 \\ D. & \quad \frac{8!}{(8-3)!3!} = 56 \end{align*} \][/tex]
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