<p>Alicia is arranging photographs of 5 family members in a row on her fridge. Her mother and father must be on opposite ends. How many possible arrangements are there?</p>
<p>Six chairs are placed in a row to seat six people. How many different seating arrangements are available if two people insist on sitting next to each other?</p>
<p>The answer for the first one is 12 and the second one is 240</p>
<p>Q1. There are 3! = 6 ways to position the remaining three members, and two ways to position the parents. 2*3! = 12.</p>
<p>Q2. We treat the two people as a “twin” and think of seating five people, four people plus the “twin.” 5! = 120. Now we distribute the “twin” which has two ways (AB or BA). 2*5! = 240.</p>
<p>First one I would use “space math”… what my math teacher calls it.</p>
<p>Basically five choices, how many possibilities are “possible”, and then multiply it out.</p>
<p>The first one has to be father or mother, so 2.
Second can be any of the 3 other members.
Third can be any of the 2 remaining members.
Fourth has to be the last member, so 1.
Fifth must be father or mother(like one) so 1 since the first one took the other.</p>
<p>2x3x2x1x1=12</p>
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<p>Second one is the same thing. With different numbers. Still space math. Look it up online if you want to learn it.</p>
<p>Why would you use 5! ?</p>
<p>Never heard of “space math” but I’ve used the technique before.</p>
<p>The 5! comes from arranging four people, plus the “twin” (the two people that insist on sitting next to each other, we count that as one “person”).</p>