IMPOSSIBLE 单词的字母有多少种排列方式可以使所有元音组合在一起？
排列被称为按顺序组织组、主体或数字的过程，从集合中选择主体或数字，被称为组合，其中数字的顺序无关紧要。
在数学中，排列也被称为组织一个群的过程，其中一个群的所有成员都被排列成某种顺序或顺序。如果组已经排列，则置换过程称为对其组件的重新定位。排列发生在几乎所有数学领域。它们大多出现在考虑某些有限集合上的不同命令时。
置换公式
在排列中，从一组 n 个事物中挑选出 r 个事物，没有任何替换。在这个挑选的顺序。
nPr = (n!)/(n – r)!
Here,
n = group size, the total number of things in the group
r = subset size, the number of things to be selected from the group
组合
组合是从集合中选择数字的函数，这样（不像排列）选择的顺序无关紧要。在较小的情况下，可以计算组合的数量。这种组合被称为一次合并n个事物而不重复。组合起来，顺序无关紧要，您可以按任何顺序选择项目。对于那些允许重复出现的组合，经常使用术语 k-selection 或 k-combination with replication。
组合配方
组合 r 个东西是从一组 n 个东西中挑选出来的，挑选的顺序无关紧要。
nCr = n!⁄((n-r)! r!)
Here,
n = Number of items in set
r = Number of things picked from the group
IMPOSSIBLE 单词的字母有多少种排列方式可以使所有元音组合在一起？
解决方案：
Vowels are: I,I,O,E
If all the vowels must come together then treat all the vowels as one super letter, next note the letter ‘S’ repeats so we’d use
7!/2! = 2520
Now count the ways the vowels in the super letter can be arranged, since there are 4 and 1 2-letter(I’i) repeat the super letter of vowels would be arranged in 12 ways i.e., (4!/2!)
= (7!/2! × 4!/2!)
= 2520(12)
= 30240 ways
类似问题
问题1：有多少种字母排列方式可以使所有元音组合在一起，单词是CORPORATION？
解决方案：
Vowels are :- O,O,A,I,O
If all the vowels must come together then treat all the vowels as one super letter, next note the R’r letter repeat so we’d use
7!/2! = 2520
Now count the ways the vowels in the super letter can be arranged, since there are 5 and 1 3-letter repeat the super letter of vowels would be arranged in 20 ways i.e., (5!/3!)
= (7!/2! × 5!/3!)
= 2520(20)
= 50400 ways
问题 2：“数学”这个词的字母有多少种不同的排列方式，使得元音必须总是在一起？
解决方案：
Vowels are :- A,A,E,I
Next, treat the block of vowels like a single letter, let’s just say V for vowel. So then we have MTHMTCSV – 8 letters, but 2 M’s and 2 T’s. So there are
8!/2!2! = 10,080
Now count the ways the vowels letter can be arranged, since there are 4 and 1 2-letter repeat the super letter of vowels would be arranged in 12 ways i.e., (4!/2!)
= (8!/2!2! × 4!/2!)
= 10,080(12)
= 120,960 ways
问题 3：RAINBOW 单词的字母有多少种排列方式，其中元音从不在一起？
解决方案：
Vowels are :- A, I, O
Consonants are:- R, N, B, W.
Arrange all the vowels in between the consonants so that they can not be together. There are 5 total places between the consonants. So, vowels can be organize in 5P3 ways and the four consonants can be organize in 4! ways.
Therefore, the total arrangements are 5P3 * 4! = 60 * 24 = 1440