<p>One boy and five girls must stand in line for a school picture. If the boy cannot stand first or last in line, how many different ways are there to arrange the children?
A) 720
B) 480
C)360
D)240
E) 120</p>
<p>You have 6 people. They have to fill spaces A, B, C, D, E, and F. Let’s say we fill in all girls where possible first:</p>
<p>For space A, you can put any of the 5 girls: 5 possibilities
For space B, you can put the boy or any of the remaining 4 girls: 5 possibilities
For space C, you can put the boy or any of the remaining 3 girls: 4 possibilities
For space D, you can put the boy or any of the remaining 2 girls: 3 possibilities
For space E, you can put only the boy: 1 possibility
For space F, you can only put the remaining girl: 1 possibility</p>
<p>Total combinations: 5<em>5</em>4<em>3</em>1*1=<a href=“C”>u</a> 300</p>
<p>Thank you!!</p>
<p>The above answer is wrong. You need to do the special cases first (well NEED is maybe too strong, but this is one way that always works). So in other words, do the first and last positions, and THEN the middle positions.</p>
<p>So the answer is 5<em>4</em>4<em>3</em>2*1 = 480.</p>
<p>^ Thanks DrSteve</p>
<p>So space A, you can put the 5 girls, (5)
space B, (the last spot) you can put the remaining 4 girls, (4)
space C, you can put a boy or the remaining 3 girls (4)
space D, you can put a boy or remaining 2 girls, (3)
space E, you can put a boy or remaining girl, (2)
space F, you can put the boy (1)
5<em>4</em>4<em>3</em>2*1= 480</p>
<p>yes, DrSteve is right… 314159265’s answer is wrong and it isn’t even in the answer choices.
you have to start with the first and the last slot first since the boy can’t stand at either ends…
I’ll just put the numbers in the correct position for clarification. </p>
<p>5 < first girl… the guy can’t be at the front. step 1
4 < the remaining girl + the guy. step 3
3 < assuming the guy is taken, then 3 possibilities (you could have assumed that a girl was taken too) step 4
2 < the remaining two girls step 5
1 < the last girl. step 6
4 < last girl… the guy can’t be at the end. step 2</p>
<p>5x4x3x2x1x4 = 480</p>
<p>Yep - you got it!</p>
<p>Holy crap; I screwed that one up. Just ignore my original post in the thread. :P</p>