An example for the combinatorial identity
Consider the set . Recall that the LHS counts the number of ways to choose elements from , or in other words the number of -sets of . At the RHS the term counts the number of -sets of that contain the element (note that "has been chosen" and we are only left to choose from the remaining ); the term counts the number of -sets of that do not contain the element (all elements are chosen from ).
Below we illustrate the identity with an example. We set and . The identity is now (or ). Sets that contain the element are colored red and sets that do not contain 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}