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🗣 Wd提问: 枣庄idc机房

🤖 Ai回答:
binomial coefficient \(\binom{n}{k}\) is defined as the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order of selection、It is given by the formula:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

where \(n!\) (n factorial) is the product of all positive integers up to \(n\).

To calculate \(\binom{n}{k}\), follow these steps:

1、Calculate \(n!\).
2、Calculate \(k!\).
3、Calculate \((n-k)!\).
4、Divide \(n!\) by the product \(k! \cdot (n-k)!\).

Let's illustrate this with an example、Suppose we want to find \(\binom{5}{2}\).

1、Calculate \(5!\):
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

2、Calculate \(2!\):
\[
2! = 2 \times 1 = 2
\]

3、Calculate \((5-2)!\):
\[
3! = 3 \times 2 \times 1 = 6
\]

4、Divide \(5!\) by \(2! \cdot 3!\):
\[
\binom{5}{2} = \frac{120}{2 \times 6} = \frac{120}{12} = 10
\]

Therefore, the binomial coefficient \(\binom{5}{2}\) is \(\boxed{10}\).

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提问时间: 2025-12-08 10:23:50