SOLUTION: How many arrangements of the letters of the word OLYMPIC are possible if the C and the L are to be together in any order? Could you please describe the thinking steps one should

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Question 769856: How many arrangements of the letters of the word OLYMPIC are possible if the C and the L are to be together in any order?
Could you please describe the thinking steps one should go through in tackling such a question?
Many thanks
Guy
Answer by ramkikk66(644) About Me  (Show Source):
You can put this solution on YOUR website!
How many arrangements of the letters of the word OLYMPIC are possible if the C and the L are to be together in any order? 
Ans:
Okay, here's one approach.
OLYMPIC has 7 letters - call them 7 units. 
Since C and L have to be together, consider them as a single unit.
So now we have 6 units to arrange. How many ways?
Think of it as putting 6 units into 6 slots.
The first slot can be filled in 6 ways, since you can choose any unit.
The 2nd slot can be filled in 5 ways, since you have only 5 units left and can 
choose any unit from that.
Similarly, 3rd slot can be filled in 4 ways and so on.
So, the 6 slots can be filled i.e. the 6 units can be arranged, in 6*5*4*3*2*1 
(aka 6! or factorial 6) ways. Which is red%28720%29 arrangements.
Now here's the catch - for each of these arrangements, the 2 letters C and L 
can be arranged in 2 ways (CL and LC) - they are 2 different arrangements, but
the two are still together so it is a valid one.
So the total number of arrangements of the 7 letters, keeping C and L 
together, is 720 * 2 = 1440.
Hope you got it :)