How many 7-digit even numbers less than 3 000 000 can be formed using all the digits1, 2, 2, 3, 5, 5, 6 ?

To solve this problem, we need to consider the following steps:

1. Since the number must be less than 3,000,000, the first digit can only be 1 or 2. So, there are 2 options for the first digit.

2. The number must be even, so the last digit can only be 2 or 6. So, there are 2 options for the last digit.

3. For the remaining 5 digits, we have 5 slots to fill and we have 5 digits (1, 2, 3, 5, 5). However, the digit 2 and 5 are repeated twice. So, we use the formula for permutations of multiset: n! / (r1! * r2! * ... * rk!) where n is the total number of elements and r1, r2, ..., rk are the number of times each distinct element is repeated. In this case, it is 5! / (2! * 2!) = 30.

4. Therefore, the total number of 7-digit even numbers less than 3,000,000 that can be formed using all the digits 1, 2, 2, 3, 5, 5, 6 is 2 (for the first digit) * 2 (for the last digit) * 30 (for the remaining 5 digits) = 120.