<ol>
<li>The digits 1,2,3,4 are to be arranged randomly to make a positive four-digit integer. What is the probability that the digits 1, 2, and 3 will be directly next to each other, in that order, from left to right?</li>
</ol>
<p>A. 1/12
B. 1/8
C.1/6
D. 1/4
E. 1/3</p>
<p>The answer is A. 1/12. Can somebody please explain why that's the
answer?</p>
<ol>
<li>At North High School, the ski club has 15 members and the debate club has 12 members. If a total of 11 students belong to only one of the two clubs, how many students belong to to both clubs?</li>
</ol>
<p>A. 2
B.7
C.8
D.12
E.16</p>
<p>The answer is C. 8. Can somebody please explain this question?</p>
<p>13) Just do 4 x 3 x 2 x 1 = 24 and that is how many numbers can be made out of 1,2,3, and 4. Then you know if 1, 2, and 3 are next to each other there can only be two numbers like that (4123 and 1234) so 2/24 = 1/12</p>
<p>19) 15 + 12 = 27 and then subtract 11 and you get 16. Since it would have to be between the two groups divide by 2 and the answer is 8</p>
<ol>
<li><p>You can see that there are only two ways for the numbers 123 to be in order: 1234 and 4123. What remains is how many total outcomes there can be. There’s a shortcut to this: you can just do 4! (four factorial) which is just 4<em>3</em>2*1=24. Therefore 2/24= 1/12. If you don’t know the trick, you can think about it this way: Say there are four blanks that you can write numbers in. </p>
<pre><code> __ __ __ __
</code></pre></li>
</ol>
<p>You can write 1,2,3,or 4 in the first blank, one of the other three in the second one, then you have 2 choices for the third blank and just one for the fourth. Again, 4<em>3</em>2*1. If all else fails, you could just list all the combinations out; this would take a long time, and if you didn’t know the theory, you’d be likely to make a mistake, but it would get the work done.</p>
<ol>
<li>The trick here (or what I did) is to draw a venn diagram. In the middle, there’s x, which is the number you’re looking for. Now, I’m going to say that the number of people ONLY in the ski club is y. That means the number of people ONLY in the debate club is 11-y. So, you can make a system of equations:</li>
</ol>
<p>x+y=15
x=(11-y)=12</p>
<p>Well, we don’t really care what y is, so just add the two equations, getting</p>
<p>2x+11=27 And solve. You should get x=8</p>
<p>Hope this helps and doesn’t confuse you!
–AdmireEuler</p>
<p>Edit: didn’t see this had already been answered. alihaq717’s answers are faster…</p>