2116. Choose - JulTob/Mathematics GitHub Wiki

🌟 "n Choose k" Explained

The expression $( \binom{n}{k} )$ , pronounced "n choose k," represents the number of ways to select k elements from a set of n distinct elements, without considering the order. It is a key concept in combinatorics.

Formula:

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

• $( n! )$: The factorial of $( n )$ , which is the product of all positive integers up to $( n )$.

n! = n \cdot (n-1) \cdot \dots \cdot 1 

• $( k! )$: The factorial of $( k )$.
• $( (n-k)! )$: The factorial of $( n-k )$.

Key Points:

1. Order Does Not Matter: It calculates combinations, not permutations.
2. Symmetry Property: $( \binom{n}{k} = \binom{n}{n-k} )$, since selecting $( k )$ elements from $( n )$ is equivalent to leaving out $( n-k )$ elements.

Example:

How many ways can you choose 2 elements from a set of 4 $( {A, B, C, D} )$ ?

\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \cdot 3}{2 \cdot 1} = 6

Possible combinations: $( {A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D} )$ .

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Intuitive Interpretation:

Imagine a group of $( n )$ people. $( \binom{n}{k} )$ answers the question: "How many ways can we form a team of $( k )$ people from this group?"

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🌈 Choosing 2 from 4 Colored Circles

Imagine you have 4 colored circles:

🔴 🟢 🔵 🟡

You want to select 2 circles from this group, regardless of order. The number of ways to do this is ( \binom{4}{2} ), which equals:

\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \cdot 3}{2 \cdot 1} = 6

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All Possible Combinations:

1. 🔴 🟢
2. 🔴 🔵
3. 🔴 🟡
4. 🟢 🔵
5. 🟢 🟡
6. 🔵 🟡

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Key Takeaways:

• Order doesn’t matter (e.g., 🔴 🟢 is the same as 🟢 🔴).
• Each pair is a unique selection, leading to 6 total combinations.