In how many ways can 8 men and 8 women stand in a line, alternating in gender?

In mathematics, permutation is known as the process of arranging a set in which all the members of a set are arranged into some series or order. The process of permuting is known as the rearranging of its components if the set is already arranged. Permutations take place, in more or less important ways, in almost every area of mathematics. They frequently appear when different commands on certain finite sets are considered.

What is a Combination?

A combination is an act of choosing items from a group, such that (not like permutation) the order of choice does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the union of n things taken k at a time without repetition  In combination you can select the items in any order. To those combinations in which re-occurrence is allowed, the terms k-selection or k-combination with replication are frequently used.

Permutation Formula

In permutation r things are selected from a set of n things without any replacement. In this order of selection matter.

ⁿP_r = (n!) / (n-r)!

Here,

n = set size, the total number of items in the set

r = subset size , the number of items to be selected from the set

Combination Formula

In combination r things are selected from a set of n things and where the order of selection does not matter.

ⁿC_r = n!/(n−r)!r!

Here, 

n = Number of items in set

r = Number of items selected from the set

In how many ways can 8 men and 8 women stand in a line, alternating in gender?

Solution:

8 men’s, 8 women’s 

1) even places by women, odd places by men

There are 8 number of ways for men standing in the first position, similarly there are 

8 no ways for women standing in second position then there are 7 number of ways for men standing in the 3rd position and there are 7 number of ways for women standing in the 4th position and so on…until all the places are filled.

So, there are 8 number of ways for men standing in the odd position and for women standing in the even position

Number of ways

⇒ 8! × 8!

⇒ (8!)²

2) even places by men, odd places by women

There are 8 number of ways for women standing in the first position, similarly there are 8 no ways for men standing in second position then there are 7 number of ways for women standing in the 3rd position and there are 7 number of ways for men standing in the 4th position and so on…until all the places are filled.

So, there are 8 number of ways for men standing in the even position and for women standing in the odd position  

Number of ways  

⇒ 8! × 8!

⇒ (8!)²

Total number of ways  

⇒ (8!)² + (8!)²

⇒ 2 × (8!)²

Similar Questions

Question 1: In how many ways can 7 men and 7 women stand in a line, but no women can stand together?

Solution:

7 men’s, 7 women’s

1) even places by women, odd places by men

There are 7 number of ways for men standing in the first position, similarly there are 7 no ways for women standing in second position then there are 6 number of ways for men standing in the 3rd position and there are 6 number of ways for women standing in the 4th position and so on…until all the places are filled.

So, there are 7 number of ways for men standing in the odd position and 7 no of ways for women standing in the even position

Number of ways  

⇒ 7! × 7!

⇒ (7!)²

2) even places by men, odd places by women 

There are 7 number of ways for women standing in the first position, similarly there are 7 no ways for men standing in second position then there are 6 number of ways for women standing in the 3rd position and there are 6 number of ways for men standing in the 4th position and so on…until all the places are filled.

So, there are 7 number of ways for women standing in the odd position and 7 no of ways for men standing in the even position 

Number of ways  

⇒ 7! × 7!

⇒ (7!)²

Total number of ways  

⇒ (7!)² + (7!)²

⇒ 2 × (7!)²