Correct Option (1)

: Sunday

To determine the day of the week for 10th October 2027, we calculate the number of odd days from a known reference point. Given that 10th October 2021 is a Sunday, we proceed as follows:

A common year consists of 365 days, which is 52 weeks and 1 odd day. Consequently, a specific date in a common year shifts forward by one day in the subsequent common year.
A leap year consists of 366 days, which is 52 weeks and 2 odd days. If February 29th is crossed when moving from one year to the next, the date shifts forward by two days.

Applying this principle year by year:

10th October 2022: The year 2021 is a common year. Thus, 10th October 2022 will be Sunday + 1 day = Monday.
10th October 2023: The year 2022 is a common year. Thus, 10th October 2023 will be Monday + 1 day = Tuesday.
10th October 2024: The year 2023 is a common year, but 2024 is a leap year. As the period from October 2023 to October 2024 includes February 29th, 2024, 10th October 2024 will be Tuesday + 2 days = Thursday.
10th October 2025: The year 2024 is a leap year, but the leap day (February 29th) has already passed when transitioning from October 2024 to October 2025. The year 2025 is a common year. Thus, 10th October 2025 will be Thursday + 1 day = Friday.
10th October 2026: The year 2025 is a common year. Thus, 10th October 2026 will be Friday + 1 day = Saturday.
10th October 2027: The year 2026 is a common year. Thus, 10th October 2027 will be Saturday + 1 day = Sunday.

Therefore, 10th October 2027 will be a Sunday.

Incorrect Options:

Options 2 (Monday), 3 (Tuesday), and 4 (Saturday) are incorrect as they do not align with the calculated day of the week for 10th October 2027. The systematic application of odd day calculations from the given reference point definitively establishes Sunday as the correct day.