The arithmetic mean of 15 numbers is 41.Then the sum of these numbers is

The correct answer is (a) 414.

The arithmetic mean of a set of numbers is the sum of the numbers divided by the number of numbers. In this case, the arithmetic mean is 41, and there are 15 numbers. Therefore, the sum of the numbers is $41 \times 15 = 414$.

Option (b) is incorrect because it is the sum of 14 numbers, not 15 numbers. Option (c) is incorrect because it is the sum of 30 numbers, not 15 numbers. Option (d) is incorrect because it is the sum of 31 numbers, not 15 numbers.