2022 AMC 8

Complete problem set with solutions and individual problem pages

Problem 8 Easy

What is the value of

\frac{1}{3}\cdot\frac{2}{4}\cdot\frac{3}{5}\cdots\frac{18}{20}\cdot\frac{19}{21}\cdot\frac{20}{22}?

  • A.

    \frac{1}{462}

  • B.

    \frac{1}{231}

  • C.

    \frac{1}{132}

  • D.

    \frac{2}{213}

  • E.

    \frac{1}{22}

Answer:B

Solution 1

Note that common factors (from 3 to 20, inclusive) of the numerator and the denominator cancel. Therefore, the original expression becomes

\frac{1}{{3}}\cdot\frac{2}{{4}}\cdot\frac{{3}}{{5}}\cdots\frac{{18}}{{20}}\cdot\frac{{19}}{21}\cdot\frac{{20}}{22}=\frac{1\cdot2}{21\cdot22}=\frac{1}{21\cdot11}=\boxed{\textbf{(B) } \frac{1}{231}}.

 

Solution 2

The original expression becomes

\frac{20!}{22!/2!} = \frac{20! \cdot 2!}{22!} = \frac{20! \cdot 2}{20! \cdot 21 \cdot 22} = \frac{2}{21 \cdot 22} = \frac{1}{21 \cdot 11} = \boxed{\textbf{(B) } \frac{1}{231}}.

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