Symmetry and Fixed Cases Principle
Lesson · Intermediate
Combinatorics
When a set of cases has a symmetry, we can often pair each case with another case that behaves in the opposite or corresponding way.
The cases that are unchanged by the symmetry are called fixed cases. These must be considered separately.
Each sign is chosen independently in
How many choices of signs make the expression positive?
There are
total choices of signs.
For every choice of signs, reverse all the signs. If the original value is
the new value is
Thus, the positive and negative values are paired.
It remains to find
Since
the numbers receiving + signs must have sum 14.
First consider the subsets containing 7. The remaining elements must have sum 7, giving
There are 3 such subsets.
For each one, its complement also has sum 14. Therefore,
Hence,
Suppose α, β, γ are complex numbers, and exactly seven distinct values can be obtained from
Prove that one of these values is 0.
The possible expressions can be paired with their negatives:
Thus every nonzero distinct value is paired with a different value, its negative.
Therefore, the number of nonzero distinct values is even.
But there are exactly seven distinct values, which is odd.
Hence one value must be fixed under negation:
Therefore, one of the seven possible values is 0.