5个元音10个辅音能组成多少个3个元音6个辅音的词？

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。

ⁿP_r= (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。

ⁿC_r = n!/((n-r)! r!)

Here,

n = Number of items in set

r = Number of things picked from the group.

5个元音10个辅音能组成多少个3个元音6个辅音的词？

回答：

Total no. of vowels = 5 

Total no. of consonants = 10

No. of words with 3 vowels and 6 consonants

3 vowels can be selected from 5 vowels = ⁵C₃ ways = n!/(n-r)!r!= 5!/(5-3)!3! =10 ways

6 consonants can be selected from 10 consonants = ¹⁰C₆ways =  n!/(n-r)!r! = 10!/(10-6)!6! = 210 ways

Total selection = ⁵C₃  ×  ¹⁰C₆

Now, 9 letters in each selection can be arranged in 9! ways

Total no. of words = ⁵C₃ ×  ¹⁰C₆  × 9!

                                      = 10  ×  210 × 9!

                            = 2100 × 9!

                            = 762,048,000 words

类似问题

问题1：如果给出5个元音和6个辅音，那么3个元音和3个辅音可以组成多少个6个字母的单词。

回答：

Total no. of vowels = 5

Total no. of consonants = 6

The no. of  6 letter words with 3 vowels and 3 consonants

3 vowels can be selected from 5 vowels = ⁵C₃ ways = n!/(n-r)!r!= 5!/(5-3)!3! =10 ways

3 consonants can be selected from 6 consonants = ⁶C₃ ways =  n!/(n-r)!r! = 6!/(6-3)!3! = 20 ways

Total selection = ⁵C₃  × ¹⁰C₆

Now, 6 letters in each selection can be arranged in 6! ways

Total no. of 6 letter words = ⁵C₃ × ⁶C₃  × 6!

                                         = 10 × 20 × 6!

                                         = 200 × 6! 

                                         = 1,44,000 words   

问题2：5个元音19个辅音能组成多少个3个元音5个辅音的词？

回答：

Total no. of vowels = 5

Total no. of consonants = 19

No. of words with 3 vowels and 5 consonants

3 vowels can be selected from 5 vowels = ⁵C₃ ways = n!/(n-r)!r!= 5!/(5-3)!3! =10 ways

5 consonants can be selected from 19 consonants = ¹⁹C₅ ways =  n!/(n-r)!r! = 19!/(19-5)!5! = 11,628 ways

Total selection = ⁵C₃  × ¹⁹C₅

Now, 8 letters in each selection can be arranged in 8! ways

Total no. of  words = ⁵C₃ × ¹⁹C₅ × 8!

                             = 10 × 11,628 × 8!

                             = 116280 × 8!

                             = 4,688,409,600 words   

问题3：5个元音17个辅音可以组成多少个2个元音3个辅音的单词？

回答：

Total no. of vowels = 5

Total no. of consonants = 17

No. of different words with 2 vowels and 3 consonants

2 vowels can be selected from 5 vowels = ⁵C₂ ways = n!/(n-r)!r!= 5!/(5-2)!2! =10 ways

3 consonants can be selected from 17 consonants = ¹⁷C₃ ways =  n!/(n-r)!r! = 17!/(17-3)!3! = 680 ways

Total selection = ⁵C₂ × ¹⁷C₃

Now, 5 letters in each selection can be arranged in 5! ways

Total no. of  words = ⁵C₂ × ¹⁷C₃ × 5!

                              = 10 × 680 × 5!

                              = 6800 × 5!

                              = 8,16,000 words     

问题4：4个元音7个辅音能组成多少个2个元音3个辅音的词？

回答：

Total no. of vowels = 4

Total no. of consonants = 7

No. of different words with 2 vowels and 3 consonants

2 vowels can be selected from 4 vowels = ⁴C₂ ways = n!/(n-r)!r!= 4!/(4-2)!2! = 6 ways

3 consonants can be selected from 7 consonants = ⁷C₃ ways =  n!/(n-r)!r! = 7!/(7-3)!3! = 35 ways

Total selection = ⁴C₂ × ⁷C₃

Now, 5 letters in each selection can be arranged in 5! ways

Total no. of  words = ⁴C₂ × ⁷C₃ × 5!

                             = 6 × 35 × 5!

                             = 210 × 5!

                             = 25,200 words