Explain verbally why n choose x is the same as n choose (n-x).
I showed why they are the same mathematically, but I am not sure how to word it. Thanks.
One interesting way would be to show it in Pascal's triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
....
notice that in any row, entries equidistant from either the left or the right are equal.
in my last row, the 28 is the 3rd from the left and the 3rd last at the right.
in row n, x and n-x are equidistant from left or right.
I understand the figure but I don't understand how I should word it to make it relevant to statistics
To understand why n choose x is the same as n choose (n-x), let's break it down conceptually.
The expression "n choose x", written as nCx, represents the number of ways we can choose x items from a set of n items without considering their order. So, nCx gives us the number of combinations we can form by selecting x items from a group of n items.
On the other hand, "n choose (n-x)", written as nC(n-x), represents the number of ways we can choose (n-x) items from a set of n items, once again without considering their order. Here, we are choosing a different number of items from the same group.
Now, let's consider the relationship between these two expressions. When we choose x items, we are essentially not choosing (n-x) items. And when we choose (n-x) items, we are essentially not choosing x items. So, if we consider all possible ways we can choose x items, we are simultaneously considering all possible ways we can choose (n-x) items. This means that the number of combinations for nCx and nC(n-x) must be the same because they represent the same set of choices, just viewed from different perspectives.
In other words, since both expressions count the same combinations, nCx = nC(n-x). This property is often referred to as the "complementary property" of binomial coefficients.
Therefore, whether mathematically or verbally, n choose x is indeed the same as n choose (n-x).