Mr. ‘X’ has three children. The birthday of the first child falls on the 5th Monday of April, that of the second one falls on the 5th Thursday of November. On which day is the birthday of his third child, which falls on 20th December ?

The correct option is Thursday.

Let us analyze the days and dates step-by-step to find the day of the week for the third child's birthday on 20th December.

Step 1: Determine the birthday of the first child in April
April has 30 days.
For a day of the week to occur 5 times in April, its date must be one of the last two days of the month (since 30 days = 4 weeks + 2 days). Therefore, the 5th occurrence of any day in April must fall on either the 29th or the 30th of April.
Since the first child's birthday falls on the 5th Monday of April, the 5th Monday must be either 29th April or 30th April.

Step 2: Determine the birthday of the second child in November
November has 30 days.
Similarly, for a day of the week to occur 5 times in November, its date must be either 29th November or 30th November.
Since the second child's birthday falls on the 5th Thursday of November, the 5th Thursday must be either 29th November or 30th November.

Step 3: Establish the calendar relation between April and November
Let us calculate the number of days between April and November.
Number of days in each month:
May: 31 days
June: 30 days
July: 31 days
August: 31 days
September: 30 days
October: 31 days
The total number of days from 30th April to 30th November is:
31 (May) + 30 (June) + 31 (July) + 31 (August) + 30 (September) + 31 (October) + 30 (November) = 214 days.
Let us find the remainder when 214 is divided by 7 to determine the shift in days:
$214=7\times 30+4$
This means 30th November is 4 days of the week ahead of 30th April.

Let us check the consistency of our choices for the 5th Monday of April (29th or 30th) and the 5th Thursday of November (29th or 30th):
Case A: If 30th April is Monday, then 30th November must be Monday + 4 days = Friday. But the 5th Thursday must be 29th or 30th November. If 30th November is Friday, then 29th November is Thursday. This perfectly matches the requirement that the 5th Thursday is 29th November.
Case B: If 29th April is Monday, then 30th April is Tuesday. Then 30th November must be Tuesday + 4 days = Saturday. This would make 29th November a Friday, meaning November does not have a 5th Thursday on either the 29th or 30th. Thus, Case B is invalid.

Therefore, we establish the following reference dates:
- 30th April is Monday (making 29th April the 5th Monday).
- 29th November is Thursday (the 5th Thursday of November).

Step 4: Find the day of the week for 20th December
We know that 29th November is a Thursday.
November has 30 days, so 30th November is a Friday.
The number of days from 30th November to 20th December is 20 days.
Let us find the number of odd days in 20 days by dividing by 7:
$20=7\times 2+6$
So, the day of the week for 20th December is 6 days after Friday.
Friday + 6 days = Thursday.

Thus, the birthday of the third child on 20th December falls on a Thursday.