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ba0cbcd0-06df-4074-98fd-c0655b4073eei %md An example for the combinatorial identity $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ ======================================================================================== Consider the set $[n]=\{1,2,\dots,n\}$. Recall that the LHS counts the number of ways to choose $k$ elements from $[n]$, or in other words the number of $k$-sets of $[n]$. At the RHS the term $\binom{n-1}{k-1}$ counts the number of $k$-sets of $[n]$ that contain the element $n$ (note that $n$ "has been chosen" and we are only left to choose $k-1$ from the remaining $n-1$); the term $\binom{n-1}{k}$ counts the number of $k$-sets of $[n]$ that do not contain the element $n$ (all $k$ elements are chosen from $[n-1]$). Below we illustrate the identity with an example. We set $n=6$ and $k=4$. The identity is now $\binom{6}{4}={\color{red}\binom{5}{3}}+{\color{blue}\binom{5}{4}}$ (or $15={\color{red}10}+{\color{blue}5}$ ). Sets that contain the element $6$ are colored red and sets that do not contain $6$ are colored blue.

An example for the combinatorial identity (nk)=(n1k1)+(n1k)\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}

Consider the set [n]={1,2,,n}[n]=\{1,2,\dots,n\}. Recall that the LHS counts the number of ways to choose kk elements from [n][n], or in other words the number of kk-sets of [n][n]. At the RHS the term (n1k1)\binom{n-1}{k-1} counts the number of kk-sets of [n][n] that contain the element nn (note that nn "has been chosen" and we are only left to choose k1k-1 from the remaining n1n-1); the term (n1k)\binom{n-1}{k} counts the number of kk-sets of [n][n] that do not contain the element nn (all kk elements are chosen from [n1][n-1]).

Below we illustrate the identity with an example. We set n=6n=6 and k=4k=4. The identity is now (64)=(53)+(54)\binom{6}{4}={\color{red}\binom{5}{3}}+{\color{blue}\binom{5}{4}} (or 15=10+515={\color{red}10}+{\color{blue}5} ). Sets that contain the element 66 are colored red and sets that do not contain 66 are colored blue.

0: {1, 2, 3, 4}
1: {1, 2, 3, 5}
2: {1, 2, 3, 6}
3: {1, 2, 4, 5}
4: {1, 2, 4, 6}
5: {1, 2, 5, 6}
6: {1, 3, 4, 5}
7: {1, 3, 4, 6}
8: {1, 3, 5, 6}
9: {1, 4, 5, 6}
10: {2, 3, 4, 5}
11: {2, 3, 4, 6}
12: {2, 3, 5, 6}
13: {2, 4, 5, 6}
14: {3, 4, 5, 6}