Step-by-step Solution:

Step 1: Elect the 4 Office Bearers
We need to select 4 specific officers (President, Vice-President, Secretary, Joint-Secretary) from 10 members.
Since the posts are distinct, the order of selection matters (Permutation).
Ways = \({}^{10}P_4 = 10 \times 9 \times 8 \times 7 = 5040\).

Step 2: Elect the Executive Committee Members
After electing the 4 officers, we have \(10 - 4 = 6\) members remaining.
We need to select 2 members for the Executive Committee.
Since these two positions are equal (order does not matter), we use Combination.
Ways = \({}^{6}C_2 = \frac{6 \times 5}{2 \times 1} = 15\).

Step 3: Total Ways
Total ways = (Ways for Office Bearers) × (Ways for Committee Members)
Total = \(5040 \times 15 = 75600\).