The given problem asks us to find the number of even factors of the expression \(2^7 \times 3^4 \times 5^3 \times 7\). To solve this, we can follow these steps:

First, find the total number of factors of the expression without the condition of being even. The formula to find the total number of factors of a number given its prime factorization is:

\((e_1+1)(e_2+1)(e_3+1)\ldots\)

where \(e_1, e_2, e_3, \ldots\) are the powers of the prime factors.

For our expression \(2^7 \times 3^4 \times 5^3 \times 7^1\), the total number of factors is calculated as follows:

\((7+1)(4+1)(3+1)(1+1) = 8 \times 5 \times 4 \times 2 = 320\)

Next, we need to find the number of factors that are even. A factor is even if it includes at least one factor of 2. This means we cannot choose the 0th power of 2 (which would result in an odd factor).

The number of odd factors can be found by excluding all instances of 2, i.e., considering only \(3^4 \times 5^3 \times 7^1\):

Total number of odd factors:

\((4+1)(3+1)(1+1) = 5 \times 4 \times 2 = 40\)

Hence, the number of even factors is given by:

\(320 - 40 = 280\)

Therefore, the number of even factors of the given expression is 280.

This confirms that the correct answer is 280, as provided in the options.