EF Exercises

Exercises 23.4 Exercises

Choose numbers

x, y

so that the equality in the Binomial Theorem becomes

\sum_{k = 0}^{n} (\binom{n}{k}) 2^{k} = 3^{n} .

Choose numbers

x, y

so that the equality in the Binomial Theorem becomes

(\binom{n}{0}) - (\binom{n}{1}) + (\binom{n}{2}) - (\binom{n}{3}) + \dots + (- 1)^{n} (\binom{n}{n}) = 0 .

The equality from Task a can be rearranged to yield

\begin{matrix} (\binom{n}{0}) + (\binom{n}{2}) + (\binom{n}{4}) + \dots + (\binom{n}{m_{1}}) \\ = (\binom{n}{1}) + (\binom{n}{3}) + (\binom{n}{5}) + \dots + (\binom{n}{m_{2}}), \end{matrix}

where

\begin{aligned} m_{1} & = {\begin{cases} n, & n even, \\ n - 1, & n odd, \end{cases} & m_{2} & = {\begin{cases} n - 1, & n even, \\ n, & n odd . \end{cases} \end{aligned}

What does this rearranged formula tell you about the subsets of a set of size

n ?

What is the sum on the left counting? What is the sum on the right counting?

Elementary Foundations: An introduction to topics in discrete mathematics