Problem: Find the order of the subgroup H, where H1 and H2 are distinct subgroups of a finite group G, each of order 2, and H is the smallest subgroup of G containing H1 and H2.

Solution:
To find the order of the subgroup H, we need to consider the group generated by H1 and H2. Let's break down the solution into smaller steps:

Step 1: Group G:
Given that G is a finite group, we can assume its order to be n. Thus, |G| = n.

Step 2: Subgroups H1 and H2:
H1 and H2 are distinct subgroups of G, each of order 2. Therefore, |H1| = |H2| = 2.

Step 3: Smallest subgroup H:
H is the smallest subgroup of G that contains H1 and H2. This means that H must contain all the elements of H1 and H2, as well as their products.

Step 4: Products of H1 and H2:
Since H1 and H2 are distinct subgroups, their products will be distinct elements in H. We can write this as H1 * H2 = {h1 * h2 | h1 ∈ H1, h2 ∈ H2}.

Step 5: Order of H:
To find the order of H, we need to count the number of distinct elements in H. Let's consider the elements of H1 and H2:

1. H1 has two elements: {e, a}, where e is the identity element and a is a non-identity element.
2. H2 also has two elements: {e, b}, where e is the identity element and b is a non-identity element.

Step 6: Distinct elements in H:
Now, let's consider the products of H1 and H2:

1. H1 * H2 = {e * e, e * b, a * e, a * b}
2. H1 * H2 = {e, b, a, ab}

Step 7: Counting the distinct elements:
Looking at the products of H1 and H2, we can see that e, b, a, and ab are all distinct elements in H. Therefore, the order of H is 4.

Conclusion:
The order of the subgroup H, which is the smallest subgroup of G containing H1 and H2, is 4.