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Prompt Engineering for Latex and 印象笔记

for latex

  1. use $$\n some formula \n$$works well,
$$
\binom{n}{m} = \frac{n!}{m!(n-m)!}
$$
(nm)=n!m!(n−m)!\binom{n}{m} = \frac{n!}{m!(n-m)!}
  1. use $ some formula $works well,
$C(n, m) = \binom{n}{m} = \frac{n!}{m!(n-m)!}$

C(n,m)=(nm)=n!m!(n−m)!C(n, m) = \binom{n}{m} = \frac{n!}{m!(n-m)!}

  1. use math \n some formula \n works well,
C(5,2)=(52)=5!2!(5−2)!=5×4×3!2×1×3!=5×42×1=10C(5, 2) = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{5 \times 4}{2 \times 1} = 10

使用通义千问来提问

You are an AI assistance. You should replace Latex [ some formula ] with math \n some formula \n, replace Latex ( another formula ) with anotherformulaanother formula. Explain Combination C(n,m) please.

结果如下:

Certainly! Here is the explanation using the specified format for mathematical formulas:

Combination C(n,m)C(n, m), also denoted as (nm)\binom{n}{m} (read as “n choose m”), represents the number of ways to choose mm elements from a set of nn elements without regard to the order of selection. This is a fundamental concept in combinatorics.

The formula for calculating combinations is:

C(n,m)=(nm)=n!m!(n−m)!C(n, m) = \binom{n}{m} = \frac{n!}{m!(n-m)!}

Where:

  • n!n! (n factorial) is the product of all positive integers up to nn.
  • m!m! (m factorial) is the product of all positive integers up to mm.
  • (n−m)!(n-m)! is the factorial of the difference between nn and mm.

For example, if you want to find how many ways there are to choose 2 elements from a set of 5 elements, you would calculate:

C(5,2)=(52)=5!2!(5−2)!=5×4×3!2×1×3!=5×42×1=10C(5, 2) = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{5 \times 4}{2 \times 1} = 10

So, there are 10 different ways to choose 2 elements from a set of 5.