2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 8 Easy

Consider the following 100 sets of 10 elements each:

\left\{ 1,2,3,\dots,10,\right\}

\left\{ 11,12,13,.,20\right\}

\left\{ 21,22,23,..,30\right\}

:

\left\{ 991,992,993,\dots,1000\right\}.

How many of these sets contain exactly two multiples of 7?

  • A.

    40

  • B.

    42

  • C.

    43

  • D.

    49

  • E.

    50

Answer:B

Note that for each group, there is at least one multiple of 7 and at most two multiples of 7.

There are \left\lfloor \frac{1000}{7}\right\rfloor=142 multiples of 7 under 1000, while there are 100 groups. Therefore, there are 142-100=42 sets contain exactly two multiples of 7.

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