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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. \(\combination{4}{4}\)
2. \(\combination{13}{5}\)
3. \(\combination{1000000}{999998}\)
4. \(\combination{7}{0}\)
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
\(1 + 2 + 3 + \dotsb + n\text{.}\)
