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22.6: Exercises

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    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.

    Exercise \(\PageIndex{1}\)

    \(C(4, 4)\)

    Exercise \(\PageIndex{2}\)

    \(C(13, 5)\)

    Exercise \(\PageIndex{3}\)

    \(C(1000000, 999998)\)

    Exercise \(\PageIndex{4}\)

    \(C(7, 0)\)

    Exercise \(\PageIndex{5}\)

    \(C(10, 6) \cdot C(6, 3)\)

    Exercise \(\PageIndex{6}\)

    \(C(10, 9) / C(5, 2)\)

    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.

    Exercise \(\PageIndex{7}\)

    \(\displaystyle C(n,k) = \dfrac{n}{k} \cdot C(n-1,k-1)\)

    Exercise \(\PageIndex{8}\)

    \(\displaystyle C(n,k) = \dfrac{n}{n - k} \cdot C(n-1,k)\)

    Exercise \(\PageIndex{9}\)

    \(\displaystyle C(n,k) = \dfrac{n - k + 1}{k} \cdot C(n,k-1)\)

    Exercise \(\PageIndex{10}\)

    \(\displaystyle C(n+k,n) = C(n+k,k)\)

    Exercise \(\PageIndex{11}\)

    Choose a value for \(m\) so that the equality in Proposition 22.4.3 becomes a formula for the sum \(1 + 2 + 3 + \cdots + n\text{.}\)

    This page titled 22.6: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.