如果总共有 23 张 CD，求 5 张 CD 的排列数

排列是一组项目的所有部分的有序排列。在排列的情况下考虑元素的出现顺序。这是一个有序的组合。

例如，一次取 r 的 n 个不同对象的排列数为ⁿ P _r ，其中；

ⁿP_r = n!/(n-r)!

= n(n−1)(n−2)…{n−(r−1)}

(Note that n! = n×(n−1)×(n−2)×…×2×1)

如果总共有 23 张 CD，请找出 5 张 CD 的排列数。

解决方案：

Given a total of 23 CDs, assuming that each CD is different, taking only 5 CDs at a time is given by; 

²³P₅ = 23!/(23 – 5)!

²³P₅ = 23!/18!

Simplifying

²³P₅ = 23 × 22 × 21 × 20 × 19 × 18!/18!

²³P₅ = 23 × 22 × 21 × 20 × 19

²³P₅ = 4037880

Therefore,

The number of permutations of 5 CDs if we have a total of 23 CDs is 4037880.

示例问题

问题 1.如果你总共有 20 名学生，求 3 名学生的排列数。

解决方案：

ⁿP_r = n!/(n-r)!

Given a total of 20 students, assuming that each student is different, take only 3 students at a time is given by;

²⁰P₃ = 20!/(20 – 3)!

²⁰P₃ = 20!/17!

Simplifying

²⁰P₃ = 20 × 19 × 18 × 17!/17!

²⁰P₃ = 20 × 19 × 18

²⁰P₃ = 6840

Therefore,

The number of permutations of 3 students if we have a total of 20 students is 6840.

问题 2. Mallika有 4 块巧克力，Mallika 想把它们送给 3 个乞丐。找出她可以通过多少种可能的方式做到这一点？

解决方案：

Here,

We have given that 

n = 4

r = 3

Therefore,

ⁿP_r = n!/(n-r)!

⁴P₃ = 4!/(4-3)!

⁴P₃ = 4!/1!

⁴P₃ = 4 × 3 × 2 × 1

⁴P₃ = 24

Hence,

Mallika can give 4 chocolates to 3 beggars in 24 ways.

问题 3. 假设一组 5 个字母 a、b、c、d、e。

找出有多少种方法可以选择不重复的 4 个字母？

解决方案：

Here,

We have given that 

n = 5

r = 4

Therefore,

ⁿP_r = n!/(n-r)!

⁵P₄ = 5!/(5-4)!

⁵P₄ = 5!/1!

⁵P₄ = 5 ×  4 × 3 × 2 × 1

⁵P₄ = 120

Hence,

We can select letters in 120 ways.

问题 4. 找出有或无意义的四个字母可以用单词“LOGARITHMS”组成，假设字母不能重复？

解决方案：

Here

We have 

n = 10

r = 4

ⁿP_r = n!/(n-r)!

¹⁰P₄ = 10!/(10-4)!

¹⁰P₄ = 10!/6!

¹⁰P₄ = 10 × 9 × 8 × 7 × 6!/6!

¹⁰P₄ = 10 × 9 × 8 × 7

¹⁰P₄ = 5040

In 5040 ways.

问题 5。如果您总共有 16 位女性，请找出 4 位女性的排列数。

解决方案：

ⁿP_r = n!/(n-r)!

Given a total of 16 women, assuming that each student is different, take only 4 women at a time is given by;

¹⁶P₄ = 16!/(16 – 4)!

¹⁶P₄ = 16!/12!

Simplifying

¹⁶P₄ = 16 × 15 × 14 × 13 × 12!/12!

¹⁶P₄ = 16 × 15 × 14 × 13

¹⁶P₄ = 43680

Therefore,

The number of permutations of 4 women if we have a total of 16 women is 43680.