THE FACTORIAL IS USEFUL FOR COUNTING THE NUMBER OF WAYS THAT A SET OF OBJECTS CAN BE ORDERED. IF A SET OF N OBJECTS IS TO BE ORDERED FROM 1ST TO NTH, THERE ARE N CHOICES FOR THE 1ST OBJECT, N-1 CHOICES FOR THE 2ND OBJECT, N-2 CHOICES FOR THE 3RD OBJECT, AND SO ON, UNTIL THERE IS ONLY 1 CHOICE FOR THE NTH OBJECT. THUS, BY THE MULTIPLICATION PRINCIPLE, THE NUMBER OF WAYS OF ORDERING THE N OBJECTS IS
N = N!
FOR EXAMPLE, THE NUMBER OF WAYS OF ORDERING THE LETTERS A, B, AND C IS 3!, OR 6:ABC, ACB, BAC, BCA, CAB, AND CBA.
THESE ORDERINGS ARE CALLED THE PERMUTATIONS OF THE LETTERS A, B, AND C.也可以用P 33表示.
例如:1, 2, 3, 4, 5这5个数字构成不同的5位数的总数为5! = 120
组合:COMBINATION
A PERMUTATION CAN BE THOUGHT OF AS A SELECTION PROCESS IN WHICH OBJECTS ARE SELECTED ONE BY ONE IN A CERTAIN ORDER. IF THESGROUPSOF SELECTION IS NOT RELEVANT AND ONLY K OBJECTS ARE TO BE SELECTED FROM A LARGER SET OF N OBJECTS, A DIFFERENT COUNTING METHOD IS EMPLOYED.
SPECIALLY CONSIDER A SET OF N OBJECTS FROM WHICH A COMPLETE SELECTION OF K OBJECTS IS TO BE MADE WITHOUT REGARD TO ORDER,SWHERES0N . THEN THE NUMBER OF POSSIBLE COMPLETE SELECTIONS OF K OBJECTS IS CALLED THE NUMBER OF COMBINATIONS OF N OBJECTS TAKEN K AT A TIME AND IS CKN.
从N个元素中任选K个元素的数目为:
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