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Exercises 22.6 Exercises
Evaluating the combination formula.
In each of
Exercises 1–6 , compute the value of the combination or formula of combinations. To obtain exact answers, you should simplify the factorial expressions before computing.
1. 2. 3.
\(\combination{1000000}{999998} \)
4. 5.
\(\combination{10}{6} \cdot \combination{6}{3} \)
6.
\(\combination{10}{9} / \combination{5}{2} \)
Combination formula identities.
In each of
Exercises 7–10 , verify the equality of combination formulas. Remember to consider the left-hand and right-hand sides of each equality
separately , manipulating/simplifying one or the other or both sides until they are the same expression.
7.
\(\displaystyle \combination{n}{k} = \frac{n}{k} \cdot \combination{n-1}{k-1} \)
8.
\(\displaystyle \combination{n}{k} = \frac{n}{n - k} \cdot \combination{n - 1}{k} \)
9.
\(\displaystyle \combination{n}{k} = \frac{n - k + 1}{k} \cdot \combination{n}{k - 1} \)
10.
\(\displaystyle \combination{n + k}{n} = \combination{n + k}{k} \)
11. Choose a value for
\(m \) so that the equality in
Proposition 22.4.4 becomes a formula for the sum
\(1 + 2 + 3 + \dotsb + n \text{.}\)
12.
Prove the identity
\begin{equation*}
\sum_{k = 0}^n \left( \combinationalt{n}{k} \cdot 2^k \right) = 3^n
\end{equation*}
by arguing that each side (separately) represents the cardinality of the set
\begin{equation*}
\setdef{ (A,B) \in \powset{X} \cartprod \powset{X} }{ A \intersection B = \emptyset }
\end{equation*}
when \(\card{X} = n \text{.}\)
