6.排列与组合

　　THERE ARE SOME USEFUL METHODS FOR COUNTING OBJECTS AND SETS OF OBJECTS WITHOUT ACTUALLY LISTING THE ELEMENTS TO BE COUNTED. THE FOLLOWING PRINCIPLE OF MULTIPLICATION IS FUNDAMENTAL TO THESE METHODS.

　　IF A FIRST OBJECT MAY BE CHOSEN IN M WAYS AND A SECOND OBJECT MAY BE CHOSEN IN N WAYS, THEN THERE ARE MN WAYS OF CHOOSING BOTH OBJECTS.

　　AS AN EXAMPLE, SUPPOSE THE OBJECTS ARE ITEMS ON A MENU. IF A MEAL CONSISTS OF ONE ENTREE AND ONE DESSERT AND THERE ARE 5 ENTREES AND 3 DESSERTS ON THE MENU, THEN 5×3 = 15 DIFFERENT MEALS CAN BE ORDERED FROM THE MENU. AS ANOTHER EXAMPLE, EACH TIME A COIN IS FLIPPED, THERE ARE TWO POSSIBLE OUTCOMES, HEADS AND TAILS. IF AN EXPERIMENT CONSISTS OF 8 CONSECUTIVE COIN FLIPS, THE EXPERIMENT HAS 28 POSSIBLE OUTCOMES,SWHERESEACH OF THESE OUTCOMES IS A LIST OF HEADS AND TAILS IN SOME ORDER.

　　☆阶乘：FACTORIAL NOTATION

　　假如一个大于1的整数N，计算N的阶乘被表示为N!，被定义为从1至N所有整数的乘积，

　　例如：4! = 4×3×2×1= 24

　　注意：0! = 1! = 1

　　☆排列：PERMUTATIONS

　　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-1) (N-2)…( 3) (2) (1) = 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.

　　　　　　　　　　　　　　　　　　　　　　　　　　PKN = N!/ (N-K)!