| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | C. 98770 |
C |
| deepseekr1 | To compute \(\dbinom{85}{82}\), we use the symmetry property of combinations: \(\dbinom{n}{k} = \dbinom{n}{n-k}\). Thus, \(\dbinom{85}{82} = \dbinom{85}{3}\). Calculating \(\dbinom{85}{3}\): \[ \dbinom{85}{3} = \frac{85 \times 84 \times 83}{3 \times 2 \times 1} \] **Numerator**: \(85 \times 84 = 7140\) \(7140 \times 83 = 592,620\) **Denominator**: \(3! = 6\) **Division**: \(592,620 \div 6 = 98,770\) Thus, \(\dbinom{85}{82} = 98,770\), corresponding to **option C**. **Answer: C** |
C |
| gpto1 | To solve the problem, we need to compute \(\dbinom{85}{82}\). First, recall the symmetry property of combinations: \(\dbinom{n}{k} = \dbinom{n}{n-k}\). This means that \(\dbinom{85}{82} = \dbinom{85}{3}\), since \(85 - 82 = 3\). Computing \(\dbinom{85}{3}\) is typically easier because it involves smaller numbers. Now, compute \(\dbinom{85}{3}\) using the combination formula: \[ \dbinom{85}{3} = \frac{85 \times 84 \times 83}{3 \times 2 \times 1} = \frac{85 \times 84 \times 83}{6} \] Calculate the numerator: \[ 85 \times 84 = 7140 \\ 7140 \times 83 = 592,620 \] Then, calculate the denominator: \[ 3 \times 2 \times 1 = 6 \] Finally, divide the numerator by the denominator: \[ \frac{592,620}{6} = 98,770 \] Looking at the options provided: - Option C is 98,770, which matches our result. **Answer:** C |
C |
| GPT-4o | To compute \(\dbinom{85}{82}\), we use the binomial coefficient formula: \[ \dbinom{n}{k} = \frac{n!}{k!(n-k)!} \] For \(\dbinom{85}{82}\), this becomes: \[ \dbinom{85}{82} = \frac{85!}{82!(85-82)!} = \frac{85!}{82! \cdot 3!} \] We can simplify this by canceling out the common \(82!\) term in the numerator and denominator: \[ \dbinom{85}{82} = \frac{85 \times 84 \times 83 \times 82!}{82! \times 3!} = \frac{85 \times 84 \times 83}{3!} \] Now we compute the factorial in the denominator: \[ 3! = 3 \times 2 \times 1 = 6 \] So, we have: \[ \dbinom{85}{82} = \frac{85 \times 84 \times 83}{6} \] Next, we calculate the numerator: \[ 85 \times 84 = 7140 \] \[ 7140 \times 83 = 592620 \] Thus: \[ \dbinom{85}{82} = \frac{592620}{6} = 98770 \] Therefore, the correct answer is: C. 98770 |
C |
| llama3.1-70B-Instruct | A Answer: $\dbinom{85}{82}=\dfrac{85\cdot84\cdot83}{3\cdot2\cdot1}=85\cdot42\cdot83/2=85\cdot 1723/2=146005/2=73002.5$ |
C |
| llama3.1-405B-Instruct | Using the formula for combinations, we have $$\dbinom{85}{82}=\frac{85!}{82!3!}=\frac{85\cdot84\cdot83\cdot82!}{82!3!}=\frac{85\cdot84\cdot83}{3\cdot2\cdot1}=\boxed{C}.$$ Final Answer: The final answer is C. I hope it is correct. |
C |
| DeepSeek-Coder-V2 | D. 101170 | C |
| Qwen2-72B-Instruct | C. 98770 | C |
| baichuan4 | B. 4680 | C |
| doubao | C. 98770. $\dbinom{n}{r}=\frac{n!}{r!(n-r)!}$. So $\dbinom{85}{82}=\frac{85!}{82!(85-82)!}=\frac{85\times84\times83}{3\times2\times1}=98770$. |
C |
| minimax | The binomial coefficient $\dbinom{n}{k}$ is defined as the number of ways to choose $k$ elements from a set of $n$ elements without regard to order. It is given by the formula: $$\dbinom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$. For the given question, we have: $$\dbinom{85}{82} = \frac{85!}{82!(85-82)!} = \frac{85!}{82! \cdot 3!}$$ Since $85! = 85 \times 84 \times 83 \times 82!$, we can simplify the expression by canceling out the $82!$ on the numerator and denominator: $$\dbinom{85}{82} = \frac{85 \times 84 \times 83}{3 \times 2 \times 1}$$ Now, we can compute the value: $$\dbinom{85}{82} = \frac{85 \times 84 \times 83}{6} = 85 \times 14 \times 83$$ $$\dbinom{85}{82} = 1190 \times 83$$ $$\dbinom{85}{82} = 98770$$ So the correct answer is: C. 98770 |
C |
| kimi | To compute \(\dbinom{85}{82}\), we can use the property of binomial coefficients that states \(\dbinom{n}{k} = \dbinom{n}{n-k}\). This means that \(\dbinom{85}{82}\) is the same as \(\dbinom{85}{3}\) because \(85 - 82 = 3\). Now, we calculate \(\dbinom{85}{3}\) using the formula for binomial coefficients: \[ \dbinom{n}{k} = \frac{n!}{k!(n-k)!} \] So, \[ \dbinom{85}{3} = \frac{85!}{3!(85-3)!} = \frac{85 \times 84 \times 83}{3 \times 2 \times 1} \] Calculating this gives: \[ \dbinom{85}{3} = \frac{85 \times 84 \times 83}{6} = \frac{571880}{6} = 95380 \] However, this result is not among the options provided. It seems there might be a mistake in the options or in the calculation. Let's check the calculation again. \[ \dbinom{85}{3} = \frac{85 \times 84 \times 83}{3 \times 2 \times 1} = \frac{571880}{6} = 95380 \] This result is still not matching any of the options. It's possible that the question intended to ask for \(\dbinom{85}{5}\) instead, which would be: \[ \dbinom{85}{5} = \frac{85 \times 84 \times 83 \times 82 \times 81}{5 \times 4 \times 3 \times 2 \times 1} = \frac{34194220}{120} = 284953 \] Again, this is not among the options. It seems there is a discrepancy in the question or the provided options. If we assume the question meant to ask for \(\dbinom{85}{3}\), then the correct answer would be 95380, which is not listed. However, if we consider the possibility of a typo and the question meant to ask for \(\dbinom{85}{5}\), then the closest answer to the options provided would be: \[ \dbinom{85}{5} = \frac{85 \times 84 \times 83 \times 82 \times 81}{5 \times 4 \times 3 \times 2 \times 1} = \frac{34194220}{120} = 284953 \] This is still not matching any of the options. If we consider the closest option to the correct calculation, it would be: \[ \boxed{C} \] But please note that the correct calculation for \(\dbinom{85}{3}\) or \(\dbinom{85}{5}\) does not match any of the options provided. |
C |
| wenxinyiyan | D. 101170 Explanation: We are asked to compute $\dbinom{85}{82}$, which is a binomial coefficient representing the number of ways to choose 82 items from 85 items. Using the formula for binomial coefficients, we have $\dbinom{n}{k} = \frac{n!}{k!(n-k)!}$ where $n!$ denotes the factorial of $n$. Applying this formula to our situation, we get $\dbinom{85}{82} = \frac{85!}{82!(85-82)!}$ $= \frac{85!}{82! \times 3!}$ $= \frac{85 \times 84 \times 83}{3 \times 2 \times 1}$ $= 101170$ |
C |
| zhipu | To compute \(\dbinom{85}{82}\), we use the binomial coefficient formula: \[ \dbinom{n}{k} = \frac{n!}{k!(n-k)!} \] For \(\dbinom{85}{82}\), \(n = 85\) and \(k = 82\). Notice that \(\dbinom{85}{82} = \dbinom{85}{3}\) because \(\dbinom{n}{k} = \dbinom{n}{n-k}\). So, we need to compute \(\dbinom{85}{3}\): \[ \dbinom{85}{3} = \frac{85!}{3!(85-3)!} = \frac{85!}{3! \cdot 82!} \] We can simplify this by canceling out the \(82!\) terms: \[ \dbinom{85}{3} = \frac{85 \times 84 \times 83}{3 \times 2 \times 1} \] Now, calculate the numerator: \[ 85 \times 84 \times 83 = 85 \times 84 = 7140 \quad \text{and then} \quad 7140 \times 83 = 592620 \] Next, calculate the denominator: \[ 3 \times 2 \times 1 = 6 \] Finally, divide the numerator by the denominator: \[ \frac{592620}{6} = 98770 \] Thus, the value of \(\dbinom{85}{82}\) is 98770. Therefore, the correct answer is: C. 98770 |
C |
| gpto1 | To solve the problem, we need to compute \(\dbinom{85}{82}\). First, recall the symmetry property of combinations: \(\dbinom{n}{k} = \dbinom{n}{n-k}\). This means that \(\dbinom{85}{82} = \dbinom{85}{3}\), since \(85 - 82 = 3\). Computing \(\dbinom{85}{3}\) is typically easier because it involves smaller numbers. Now, compute \(\dbinom{85}{3}\) using the combination formula: \[ \dbinom{85}{3} = \frac{85 \times 84 \times 83}{3 \times 2 \times 1} = \frac{85 \times 84 \times 83}{6} \] Calculate the numerator: \[ 85 \times 84 = 7140 \\ 7140 \times 83 = 592,620 \] Then, calculate the denominator: \[ 3 \times 2 \times 1 = 6 \] Finally, divide the numerator by the denominator: \[ \frac{592,620}{6} = 98,770 \] Looking at the options provided: - Option C is 98,770, which matches our result. **Answer:** C |
C |
| deepseekv3 | To compute the binomial coefficient \(\dbinom{85}{82}\), we can use the property of combinations that states: \[ \dbinom{n}{k} = \dbinom{n}{n - k} \] Applying this property: \[ \dbinom{85}{82} = \dbinom{85}{3} \] Next, we use the formula for combinations: \[ \dbinom{n}{k} = \frac{n!}{k!(n - k)!} \] Substituting \(n = 85\) and \(k = 3\): \[ \dbinom{85}{3} = \frac{85!}{3!(85 - 3)!} = \frac{85!}{3! \cdot 82!} \] The \(82!\) terms cancel out: \[ \dbinom{85}{3} = \frac{85 \times 84 \times 83 \times 82!}{3! \times 82!} = \frac{85 \times 84 \times 83}{3!} \] Since \(3! = 6\): \[ \dbinom{85}{3} = \frac{85 \times 84 \times 83}{6} \] Calculating the numerator: \[ 85 \times 84 = 7140 \] \[ 7140 \times 83 = 592, 620 \] Dividing by 6: \[ \frac{592, 620}{6} = 98, 770 \] Therefore, the value of \(\dbinom{85}{82}\) is \(98, 770\). The correct answer is: \boxed{C} |
C |