Symmetry (Meta) Math Example 2

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Example 2

medium
Use symmetry to evaluate ∑k=0n(nk)(−1)k\displaystyle\sum_{k=0}^{n} \binom{n}{k}(-1)^k for even n=4n=4.

Solution

  1. 1
    Write out for n=4n=4: (40)−(41)+(42)−(43)+(44)=1−4+6−4+1\binom{4}{0}-\binom{4}{1}+\binom{4}{2}-\binom{4}{3}+\binom{4}{4} = 1-4+6-4+1.
  2. 2
    Symmetry: terms pair up: (nk)\binom{n}{k} and (nn−k)\binom{n}{n-k} are equal (symmetry of binomial coefficients). With the alternating signs, term kk and term n−kn-k cancel for odd nn.
  3. 3
    For n=4n=4: 1−4+6−4+1=01-4+6-4+1 = 0. This equals (1−1)4=04=0(1-1)^4 = 0^4 = 0 by the binomial theorem.

Answer

∑k=04(4k)(−1)k=0\sum_{k=0}^{4}\binom{4}{k}(-1)^k = 0
Symmetry of the binomial coefficients ((nk)=(nn−k)\binom{n}{k}=\binom{n}{n-k}) combined with the alternating signs causes cancellation. Recognising the binomial theorem shortcut =(1−1)n=(1-1)^n gives the answer instantly.

About Symmetry (Meta)

A property of a mathematical object that remains unchanged under a specified transformation — reflection, rotation, translation, or algebraic substitution.

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More Symmetry (Meta) Examples

Example 1 easy

Show that the equation [formula] is symmetric about both coordinate axes and the origin. Verify by s

Example 3 easy

Determine whether [formula] is odd, even, or neither, by testing the symmetry condition.

Example 4 medium

Use the symmetry of [formula] (odd function) and [formula] (even function) to simplify: [formula].

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