<p>M&M plain candies come in 6 colors: brown, green, orange, red, tan, and yellow. Assume there are at least 3 of each color. If you pick three colors from a bag, how many color possibilities are there?</p>
<p>a) 18
b) 20
c) 120
d) 216
e) 729</p>
<p>The answer is D), but I somehow don't get why. I'm not sure if it is an error or not, but the explanation says you have to multiply 6<em>6</em>6= 216. </p>
<p>I thought you can do it like this:</p>
<p>6!/((6-3)!*3! ), and then multiply it by 3 since
- There are 3 slots for 6 different colors, and order does not matter
- You multiply all of this by 3 to recognize that there are 3 of the same color.</p>
<hr>
<p>pg 140</p>
<p>In a plane there are 8 points, no three of which are collinear. How many lines do the points determine?</p>
<p>A) 7
B) 16
C) 28
D) 36
E) 64</p>
<p>The answer is C, but I don't really understand the phrase collinear. Would this mean that only 5 points are colinear (only 5 points have the chance to be touched by a line which touches one of the other 5 points)?</p>
<p>I'm really bad with combinations but I can help on the second one</p>
<p>Any two points are collinear because you can draw a line through any two points. Basically what the problem is saying is that no 3 points lie on a line. the only lines that can be determined are any combination of two points. IF 3 could be collinear, then there would be fewer distinct lines because two of the lines would be the same (ex. A,B, and C are the three points. If they are not collinear you can draw line AB AC or BC but if they were all on one line, AB, AC, and BC would be the same line.)</p>
<p>For the first, you should use 6!/((6-1)!<em>1!) = 6, or 6P1 instead. You have to take one item out of six for each slot, not 3 out of 6. This gives you the number of possibilties for each slot, 6. Multiply the number of possibilities for each slot together, and you will get 6</em>6*6. </p>
<p>Recognizing that there are six choices for each slot is a much simpler and straightforward way to solve it.</p>
<p>For #2, I have a more mental approach to solving it. Between any two given points there is a line. Since there are no three points that share a line (collinear points share a line), each point will be connected with the others by one line. So, point #1 will be connected to the other 7 by 7 unique lines. Point #2 will be connected to the other 6 by 6 unique lines (the line between #1 and #2 already counted). Point #3 will be connected to the other 5 by 5 unique lines. Ect. So you get
7+6+5+4+3+2+1=28</p>
<p>You could use the formula but i personally hate combos and perms</p>
<p>M&Ms.
I could not find this question in my Barron's to verify its wording.
@mAnshin - what edition is yours?</p>
<p>I have an issue with the answer 216.
To me a group of 1 green, 1 red, and 1 orange candies is the same as a group of 1 green, 1 orange, and 1 red ones.</p>
<p>There are 6<em>6</em>6=216 possible ordered color arrangements. Since the order in a group of 3 candies is not important, this number needs to be divided by 3P1=3! (the number of all possibles "shuffles" of a given set of 3 colors in three slots: 3 choices for the first slot, 2 for the second, 1 for the third, 3<em>2</em>1=6).
My answer is 216/6=36.
Am I missing something here?</p>
<p>@gfc101
You are forgetting that although for most combinations, there will be 6 different "shuffles" but for some combinations such as red- red-red, there was only 1 "shuffle" so you cannot divide the entire thing by 6.</p>
<p>If order is not an issue, then the correct formula is
(n + k - 1)! / (k!(n-1)!)
where n is the number of different colors and k is the number of m&ms taken.</p>
<p>^of course! combinations with repetition. Thanks, khoitrinh!
If all three picked candies had to be different colors, then eliminating of "shufles" would work:
6<em>5</em>4 / (3!) = 20.
If exactly two candies out of three are the same color, there are 6 choices for that color and 5 color choices for the third candy, 6*5=30 total.
There are 6 triplets of the same color.
All together now... 20+30+6 = 56.</p>
<p>You have 3 items to choose out of the bag. There are 6 colors you can choose on the first draw, 6 the second draw, and still 6 the third draw, which gives you 6<em>6</em>6, or 216 combinations.</p>