Question 117767
I'll do the first one to show you 2 ways you can tackle these types of problems



Method #1:


Since 2 only places to put them, you can visualize of the problem as a series of slots like this:

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Now since you have 5 distinct letters, you have 5 choices for the first slot. Once you take away one letter, you are left with 4 letters. So you'll have 4 choices for the 2nd slot. Now simply multiply these two numbers to get


5*4=20


So you'll have 20 unique combinations


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Method #2:



Since we're talking about permutations, we can use the permutations formula:




*[Tex \LARGE \textrm{_{n}P_{r}=]{{{n!/(n-r)!}}} Start with the given formula




*[Tex \LARGE \textrm{_{5}P_{2}=]{{{5!/(5-2)!}}} Plug in {{{n=5}}} (this is the total number of letters that you have) and {{{r=2}}} (this is the number of letters that you can use at one time) note: the value of n is greater than r. In other words n>r.




*[Tex \LARGE \textrm{_{5}P_{2}=]{{{5!/3!}}} Subtract {{{5-2}}} to get 3




Expand 5!
*[Tex \LARGE \textrm{_{5}P_{2}=]{{{(5*4*3*2*1)/3!}}}




Expand 3!
*[Tex \LARGE \textrm{_{5}P_{2}=]{{{(5*4*3*2*1)/(3*2*1)}}}




*[Tex \LARGE \textrm{_{5}P_{2}=]{{{(5*4*cross(3*2*1))/(cross(3*2*1))}}}  Cancel




*[Tex \LARGE \textrm{_{5}P_{2}=]{{{5*4}}}  Simplify





*[Tex \LARGE \textrm{_{5}P_{2}=]{{{20}}}  Now multiply 5*4 to get 20



So 5 choose 2 (where order does matter) yields 20 unique combinations




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So no matter what method you choose, you'll get the same answer.