Question

In how many ways can we stack n different coins so that two particular coins are not adjacent to each other? [ note that m! = (1)(2)(3)(m) ]

Answer 1

Answer:

Total no. of ways they are not together = (n - 2)(n - 1)!

Step-by-step explanation:

In the question,

Total number of coins we have = n

We know that number of ways of arranging 'n' different things is given by,

n!

So,

Now, we have 2 different things which needed not to be adjacent to each other.

So,

taking those 2 things as one.

Number of ways of arranging those (n - 1) objects will be,

(n - 1)!

Now, number of ways in which the two will always be adjacent to each other are,

2(n - 1)! ways.

So,

Also,

Total number of ways = Total number of ways in which two coins are together + Total no. of ways they are not together.

So,

Total no. of ways they are not together = Total number of ways - Total number of ways in which two coins are together

So,

Total number of ways are = n!

Total no. of ways they are not together = n! - 2(n - 1)! = n(n - 1)! - 2(n - 1)! = (n - 2)(n - 1)!