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.