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<html data-require="math math-format word-problems">
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<title>Counting 2</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
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<div class="exercise">
<div class="problems">
<div id="every-n">
<div class="vars">
<var id="INCR">randRange( 3, 5 )</var>
<var id="START">randRange( 20, 50 )</var>
<var id="NTH">randRange( 7, 15 )</var>
<var id="ANSWER">START + ( NTH - 1 ) * INCR</var>
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<p data-if="isMale(1)"><var>person( 1 )</var> starts counting at <var>START</var>, and he counts by <var>cardinalThrough20( INCR )</var>s.</p><p data-else=""><var>person( 1 )</var> starts counting at <var>START</var>, and she counts by <var>cardinalThrough20( INCR )</var>s.</p>
<p class="question" data-if="isMale(1)">If <var>START</var> is the 1st number that <var>person( 1 )</var> counts. what is the <var>NTH</var>th number that he counts?</p><p class="question" data-else="">If <var>START</var> is the 1st number that <var>person( 1 )</var> counts. what is the <var>NTH</var>th number that she counts?</p>
<p class="solution" data-forms="integer"><var>ANSWER</var></p>
<div class="hints">
<p data-if="isMale(1)">What is the first number that he counts?</p><p data-else="">What is the first number that she counts?</p>
<p><code><var>START</var></code></p>
<p data-if="isMale(1)">What is the second number that he counts?</p><p data-else="">What is the second number that she counts?</p>
<p><code>\begin{align*}&amp;<var>START</var> + <var>INCR</var> \\
&amp;= <var>START + INCR</var>\end{align*}</code></p>
<p data-if="isMale(1)">What is the third number that he counts?</p><p data-else="">What is the third number that she counts?</p>
<p><code>\begin{align*}&amp;<var>START</var> + <var>INCR</var> + <var>INCR</var> \\
&amp;= <var>START</var> + (2 \times <var>INCR</var>) \\
&amp;= <var>START + 2 * INCR</var>\end{align*}</code></p>
<p data-if="isMale(1)">What is the <var>NTH</var>th number that he counts?</p><p data-else="">What is the <var>NTH</var>th number that she counts?</p>
<p><code>\begin{align*}&amp;<var>START</var> + (<var>NTH - 1</var>\times<var>INCR</var>) \\
&amp;= <var>START</var> + <var>( NTH - 1 ) * INCR</var> \\
&amp;= <var>ANSWER</var>\end{align*}</code></p>
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<div id="handshakes">
<div class="vars">
<var id="NUM">randRange( 5, 10 )</var>
<var id="ANSWER">( NUM * ( NUM - 1 ) ) / 2</var>
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<p>There are <var>NUM</var> people in a room.</p>
<p class="question">If everyone shakes everyone else's hand exactly once, how many handshakes occur?</p>
<p class="solution" data-forms="integer"><var>ANSWER</var></p>
<div class="hints">
<p>Given <var>NUM</var> people, each person shakes the hands of <var>NUM - 1</var> other people.</p>
<div>
<p>The following is almost the answer.</p>
<p><code><var>NUM</var> \times <var>NUM - 1</var> = <var>NUM * ( NUM - 1 )</var></code></p>
</div>
<p>We have double counted the handshakes though, since <var>person( 1 )</var> shaking <var>person( 2 )</var>'s hand is the same handshake as <var>person( 2 )</var> shaking <var>person( 1 )</var>'s hand.</p>
<div>
<p>Therefore, the following is the correct answer.</p>
<p><code>\dfrac{<var>NUM</var> \times <var>NUM - 1</var>}{2} = <var>ANSWER</var></code></p>
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<div id="potted-plants">
<div class="vars">
<var id="NUM_COLORS, NUM_POTS">randRangeUnique( 2, 6, 2 )</var>
<var id="ANSWER">NUM_COLORS * NUM_POTS</var>
</div>
<p data-if="isMale(1)"><var>person( 1 )</var> wants to give his friend a potted plant. At the local florist, the flowers come in <var>NUM_COLORS</var> colors, and there are <var>NUM_POTS</var> types of flower pots.</p><p data-else=""><var>person( 1 )</var> wants to give her friend a potted plant. At the local florist, the flowers come in <var>NUM_COLORS</var> colors, and there are <var>NUM_POTS</var> types of flower pots.</p>
<p class="question" data-if="isMale(1)">If he can choose any flower and any pot, how many different potted plants can <var>person( 1 )</var> buy?</p><p class="question" data-else="">If she can choose any flower and any pot, how many different potted plants can <var>person( 1 )</var> buy?</p>
<p class="solution" data-forms="integer"><var>ANSWER</var></p>
<div class="hints">
<p>If <var>person( 1 )</var> decided on a flower color, how many different potted plant combinations are there?</p>
<p data-if="isMale(1)">He can choose one of <var>NUM_POTS</var> flower pots, and so there are <var>NUM_POTS</var> different potted plants possible (given that he already chose a flower color).</p><p data-else="">She can choose one of <var>NUM_POTS</var> flower pots, and so there are <var>NUM_POTS</var> different potted plants possible (given that she already chose a flower color).</p>
<p>Since there are <var>NUM_COLORS</var> flower colors, there are <code><var>NUM_COLORS</var> \times <var>NUM_POTS</var> = <var>ANSWER</var></code> possible potted plants.</p>
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