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HOW MANY COLOURS YOU CAN CREATE WITH JUST 3 COLOURS

 

   
 Say, you are given  seven cakes  of water colour - as in rainbow.  You can select any 3 colours from the array of 7 in fixed quantity.  mix them.  You will get a new shade of colour.  How many colours can you create by choosing any 3 from 7 colours.  Here, the math branch of 'permutations and combinations' will help as to calculate the number of colour shades you can make. 
 What exactly permutations.  Suppose you have 3 elements named A, Band C.  How many ways you can arrange them.

   For first positions, there are 3 choices.
A              B                  C

 For second position there are 2 choices for each first choice.
AB and AC; BC and BA; CA and CB

 For third position, there are only one choice for each combination.
ABC and ACB; BCA and BAC; CAB and CBA;

 finally, we get six arrangements.  We also get 6 by multiplying 3*2*1, the number of choices for each position.
3*2*1 is called 'factorial 3' or by symbol 3!.

      Hence, if there are n items, n! gives the number of ways you can arrange them. or permutations of n.  The mathematical operation 'factorial' is the heart of permutations and combinations.  The operation is highly suitable for computers to do.

 Another one:  There are eight persons take part in 100 m dash(race).  Here, first three positions will be occupied by any 3 persons out of 8.  How many ways the first 3 positions will be taken up by 8 persons?  Here, we are not concerned with permutations of all the 8 persons but only the first three.  What to do?  Take factorial 8 and divide it by factorial 5 because we are not bothered about the arrangement of 5 persons who fall behind.  But the order and arrangement of first 3 is very important.
Permutations of first 3 =8!/5!=8.7.6.5.4.3.2.1/5.4.3.2.1.
            =8.7.6=336
Finally, there are 336 ways, the first 3 positions will be occupied by 8 persons.

      Coming to our initial problem, -How many colours?  You can take any 3 colours out of 7 and mix them.  In mixing , no order or sequence is important.  Hence we are going to divide 7!by 4! (since the order of 4 left out colours is not required) and also by 3!.  (since, in mixing, the order just vanish).
No.of colours =7!/4!*3! = 7.6.5.4.3.2.1/(4.3.2.1)*(3.2.1)
   = 5040/144
  = 35 colours.
    I hope we got into the soul of permutations and combinations.
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For the learners:
Permutations:  No. of ways, possibilities,choices and arrangement taking into account the order.
Combinations:  No. of ways and arrangements ignoring order

Permutations of selected set=
Factorial of full set/factorial of left out set

Combinations of      = Factorial of full set/
selected set              (Factorial of left out set)*(factorial of selected set).    

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