<ol>
<li>there are 6 red, 6 brown, 6 yellow, and 6 gray scarves packed in 24 identical, unmarked boxes, 1 scarf per box. what is the least number of boxes that must be selected in order to be sure that among the boxes selected 3 or more contain scarves of the same color?</li>
</ol>
<p>the answer is 9 but why</p>
<p>2-</p>
<p>number of throws/ number of people</p>
<pre><code> 1 7
2 6
3 6
4 4
5 2
</code></pre>
<p>in a certain game, each person threw a beanbag at a target untill the person missed the target. the table shows the results for the 25 people who played the game. for example 4 people hit the target on their first 3 throws and missed on their 4th throw. based on the info. which must be true</p>
<p>I.more than half the people hit the target on their first throw.</p>
<p>II.for all the throws attempted, more hit the target than missed the target</p>
<p>III.no one hit the target 5 times </p>
<p>the answer is I,II,III</p>
<p>please explain with a fast technique</p>
<ol>
<li><p>There are 4 different colors, so when you select 9 scarves it’s impossible to not have at least 3 of one color. With 8 scarves, you could have 2 of each, but with the 9th you must, no matter what, have at least 3 of one color.</p></li>
<li><p>I. Only 7/25 people missed their first throw, so this must be true.
III. Because the table only goes up to 5 throws, evidently nobody made it past the 5th throw.
II. This one is a little bit trickier. You know that there were 25 misses, because the chart displays the throws of 25 people, and as you can see, 7 missed on their first throw, 6 on their second, etc. 25 misses. To figure out how many hits were on target, you look at how many missed on what number. For example, 2 people missed on their 5th throw, which means that they hit their 1st, 2nd, 3rd and 4th throws. An equation I would use here is (number of people)(number of throws - 1) = number of hits by that group. => (2)(4)+(4)(3)+(6)(2)+(6)(1)+(7)(0) = 8+12+12+6+0 = 38, which is greater than 25. If you are confused by how I arrived at that equation, feel free to ask for further clarification.</p></li>
</ol>
<ol>
<li>Is more logical: I. It did say that they had the first three throw. And while there are 25 it is likely that most could have gotten it on the 1st throw. II. Notice how the 4 people hit there targets 3 to 1. which is equal to 12/4. so proportionality the number of hit is more like to be greater since more people got it on there first throw. III. Nothing in the problem indicate that someone hit the target 5 times.</li>
</ol>
<p>I think I interpreted this wrong, but for III, I solved the problem like this:</p>
<p>T H M
1 22 7
2 16 6
3 10 6
4 6 4
5 4 2</p>
<p>Assuming 25 people play, and the game ends when they miss, there are, in fact, several (4) people who would not have missed after the 5th try.</p>