Number of ways to go from Calcutta to Delhi and return by a different train:

To find the number of ways in which a person can go from Calcutta to Delhi and return by a different train, we need to consider the two legs of the journey separately.

Going from Calcutta to Delhi:

There are 10 trains plying between Calcutta and Delhi. To calculate the number of ways to go from Calcutta to Delhi, we can use the concept of combinations. Since the person can choose any one of the 10 trains, the number of ways to go from Calcutta to Delhi is equal to the number of combinations of 10 trains taken 1 at a time.

The formula for combinations is given by: nCr = n! / (r! * (n-r)!)

Using this formula, we can calculate the number of ways to go from Calcutta to Delhi as follows:

10C1 = 10! / (1! * (10-1)!) = 10

Therefore, there are 10 ways to go from Calcutta to Delhi.

Returning from Delhi to Calcutta:

Since the person has to return by a different train, they cannot take the same train they used for the journey from Calcutta to Delhi. Therefore, for the return journey, there are only 9 trains available to choose from.

Using the same formula for combinations, we can calculate the number of ways to return from Delhi to Calcutta as follows:

9C1 = 9! / (1! * (9-1)!) = 9

Therefore, there are 9 ways to return from Delhi to Calcutta.

Total number of ways:

To find the total number of ways to go from Calcutta to Delhi and return by a different train, we multiply the number of ways for each leg of the journey:

Total number of ways = Number of ways to go from Calcutta to Delhi * Number of ways to return from Delhi to Calcutta

Total number of ways = 10 * 9 = 90

Therefore, there are 90 ways in which a person can go from Calcutta to Delhi and return by a different train.