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.
\(C(4, 4)\)
\(C(13, 5)\)
\(C(1000000, 999998)\)
\(C(7, 0)\)
\(C(10, 6) \cdot C(6, 3)\)
\(C(10, 9) / C(5, 2)\)
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.
\(\displaystyle C(n,k) = \dfrac{n}{k} \cdot C(n-1,k-1)\)
\(\displaystyle C(n,k) = \dfrac{n}{n - k} \cdot C(n-1,k)\)
\(\displaystyle C(n,k) = \dfrac{n - k + 1}{k} \cdot C(n,k-1)\)
\(\displaystyle C(n+k,n) = C(n+k,k)\)
Choose a value for \(m\) so that the equality in Proposition 22.4.3 becomes a formula for the sum \(1 + 2 + 3 + \cdots + n\text{.}\)