A single plane is defined by 2 perpendicular lines. Therefore, If you have n-dimensional space and thus n mutually perpendicular lines, you can make "n choose 2" (a binomial coefficient) different mutually perpendicular planes, which is equivalent to the (n-1)th triangular number, or to (n-1)n/2. It is for this reason that there is one plane in 2-d space, 3 planes in 3-d space, 6 in 4-d space, 10 in 5-d space, and so on. Might seem counterintuitive at first, but not after you invoke binomial coefficients!

A single plane is defined by 2 perpendicular lines. Therefore, If you have n-dimensional space and thus n mutually perpendicular lines, you can make "n choose 2" (a binomial coefficient) different mutually perpendicular planes, which is equivalent to the (n-1)th triangular number, or to (n-1)n/2. It is for this reason that there is one plane in 2-d space, 3 planes in 3-d space, 6 in 4-d space, 10 in 5-d space, and so on. Might seem counterintuitive at first, but not after you invoke binomial coefficients!