<p>There are different cards in the original version, I will use Letters...</p>
<p>A B X C D</p>
<p>If the 5 letters above are placed in a row so that "X" is never at either end, how many different arrangements are possible?</p>
<p>(The right answer oughts to be 72) Why?</p>
<p>I mean, I know that 72 must be the right answer but I just cant figure out a quicker way to solve this than drawing a tree with all the possible arrangements at the end of its branches...you know what I mean</p>
<p>X can have only 3 positions. For each of the three positions of X, the number of possible arrangements is 4!=24. Thus, the total would be 3*24=72.</p>
<p>ABXCD
okay first let's consider a scenario which breaks the instruction;that is let's solve the number of arrangements where X must be at either end.If we put X at the first end we will have 24 arrangemenst.similarly,if we put it at the other end we will have 24 arrangements.Thus we will have 48 arrangements in which X is at the either end.Now the permutation of ABXCD without any restrictions is 120(5!).</p>
<p>We are interested in getting the no. of arrangements in which X is not at the either end.To get this no. we can just substract 48 from 120 which is 72.</p>
<p>Kudos to you guys.</p>
<p>To simplify what tetrisfan said:</p>
<p>You can make 6 different arrangements with A as the first letter when X is place in one position.</p>
<p>eg.
A B C X D
A B D X C
A C B X D
A C D X B
A D C X B
A D B X C</p>
<p>Therefore since X has 3 positions, you can make 3 x 6 = 18 different arrangements with A being the first letter.</p>
<p>You can do the same with B, C or D being the first letter. </p>
<p>So Place A as the first letter you get 18 different arrangements
Place B as the first letter you get 18 different arrangements
Place C as the first letter you get 18 different arrangements
Place D as the first letter you get18 different arrangements </p>
<p>So add all the different arrangements 18 + 18 + 18 + 18 = 72</p>
<p>I think I saw this question from one of my SAT prep books... Kaplan, I think.</p>
<p>those questions about possibilities always make me sick.</p>
<p>I did 5! - 2 x 4! = 72</p>