The Special Value Method Basic

Lecture Notes

The special value method has wide application beyond combinatorial identities. In the context of combinatorial identities, it is most used with binomial expansion. For example, setting $a=b=1$ in the expanded form of $(a+b)^n$: $$(a+b)^n ={n\choose 0}a^n + {n\choose 1}a^{n-1}b+{n\choose 2}a^{n-2}b^2+\cdots+{n\choose n}b^n$$ immediately gives $$2^n={n\choose 0} + {n\choose 1}+{n\choose 2}+\cdots+{n\choose n}$$



Find the value of $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +(-1)^n\binom{n}{n}$$


Calculate the value of $$\displaystyle\sum_{k=0}^{n}\frac{1}{2^k}\binom{n}{k}$$

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Note: the special value method is a generally useful technique with wide application. The "more practice problems" link will include some non-combinatorial problems.