A house has a number which needs to be identified. The following three statements are given that can help in identifying the house number.

i. If the house number is a multiple of 3, then it is a number from 50 to 59.

ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.

iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.

What is the house number?

To find the correct house number, we need to evaluate the given options against each of the three conditional statements. A conditional statement of the form "If A, then B" is true if either A is false (making it vacuously true) or both A and B are true. It is only violated (false) if A is true but B is false.

Let us review the three rules:
Statement i: If the house number is a multiple of 3, then it must be in the range [50, 59].
Statement ii: If the house number is NOT a multiple of 4, then it must be in the range [60, 69].
Statement iii: If the house number is NOT a multiple of 6, then it must be in the range [70, 79].

Let's test each option one by one:

1. Testing 54:
- 54 is a multiple of 3 (since $3\times 18=54$). It lies in the range [50, 59], so Statement i is satisfied.
- 54 is NOT a multiple of 4. Therefore, it must lie in the range [60, 69] according to Statement ii. However, 54 is not in this range, which violates Statement ii. Thus, 54 cannot be the house number.

2. Testing 65:
- 65 is NOT a multiple of 3, so Statement i is satisfied.
- 65 is NOT a multiple of 4, and it lies in the range [60, 69], so Statement ii is satisfied.
- 65 is NOT a multiple of 6. Therefore, it must lie in the range [70, 79] according to Statement iii. Since 65 is not in this range, Statement iii is violated. Thus, 65 cannot be the house number.

3. Testing 66:
- 66 is a multiple of 3 (since $3\times 22=66$). Thus, it must lie in the range [50, 59] according to Statement i. Since 66 is not in this range, Statement i is violated. Thus, 66 cannot be the house number.

4. Testing 76:
- 76 is NOT a multiple of 3 (since $76=3\times 25+1$). Since it is not a multiple of 3, Statement i is satisfied.
- 76 IS a multiple of 4 (since $4\times 19=76$). Since the condition "NOT a multiple of 4" is false, Statement ii is automatically satisfied.
- 76 is NOT a multiple of 6 (since $76=6\times 12+4$). Therefore, it must lie in the range [70, 79] according to Statement iii. Since 76 is in the range [70, 79], Statement iii is satisfied.

Since 76 is the only number that satisfies all three conditions, the house number is 76.