With combinations and permutations we are trying to determine the number of arrangements that can be made from a set of items dividing them into specific numbers or groups.
In permutations, the order of items is all that is important ; we count x, y, z as different from y, z, x. But in combinations we are concerned only that x, y and z have been selected, regardless of order; x, y, z and y, z, x are the same combination. There are thus more permutations than combinations.
n_C_x represents the number of combinations of n items taken x at a time.
n_P_x represents the number of permutations of n items taken x at a time.
How are combinations and permutations related? In general, n_C_x = n_P_x / x! (x factorial). For example:-
The number of Permutations of 5 items taken 3 at a time can be written as :-
5_P_3= 5! / (5-3)! = 5!/2! = 5*4*3*2*1/(2*1)=60
The number of Combinations of 5 items taken 3 at a time can be written as :-
5_C_3 = 5_P_3 / 3!
= 5! / (5-3)! / 3!
= 5! / (2! * 3!)
= 5*4*3*2*1 /(2*1*3*2*1) = 5*4 / 2*1 = 10
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