Pregunta
Evaluate the expression without using a calculator. \( \frac{7!}{4!(7-4)!} \) The answer is \( \square \).
Ask by Garrett Hobbs. in the United States
Nov 10,2024
Solución de inteligencia artificial de Upstudy
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The value is 35.
Solución
Calculate or simplify the expression \( 7!/(4!(7-4)!); \)
Simplify the expression by following steps:
- step0: Solution:
\(\frac{7!}{\left(4!\times \left(7-4\right)!\right)}\)
- step1: Remove the parentheses:
\(\frac{7!}{4!\times \left(7-4\right)!}\)
- step2: Subtract the numbers:
\(\frac{7!}{4!\times 3!}\)
- step3: Expand the expression:
\(\frac{7\times 6\times 5\times 4!}{4!\times 3!}\)
- step4: Simplify:
\(\frac{7\times 6\times 5}{3!}\)
- step5: Multiply the terms:
\(\frac{210}{3!}\)
- step6: Calculate the value:
\(\frac{210}{6}\)
- step7: Reduce the fraction:
\(35\)
The value of the expression \( \frac{7!}{4!(7-4)!} \) is 35.
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Simplifique esta solución The Deep Dive
To evaluate the expression \( \frac{7!}{4!(7-4)!} \), we first recognize that this is the formula for combinations, specifically \( \binom{7}{4} \). Calculating step by step, we find: 1. \( 7! = 7 \times 6 \times 5 \times 4! \) 2. Substitute this into the expression: \( \frac{7 \times 6 \times 5 \times 4!}{4! \times (3!)} \) (since \( 7 - 4 = 3 \)) 3. The \( 4! \) cancels out: \( \frac{7 \times 6 \times 5}{3!} \) 4. Since \( 3! = 6 \), this simplifies to: \( \frac{7 \times 6 \times 5}{6} = 7 \times 5 = 35 \). Therefore, the answer is \( 35 \).
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