[THOUGHT] To solve this problem, we need to count the number of seven-digit numbers formed from the digits 0, 1, and 2 where at least one of the digits 0 or 1 appears exactly twice. Here's the step-by-step approach:

1. Calculate total valid 7-digit numbers:

   • The first digit cannot be 0, so we have 2 choices (1 or 2).
   • Each of the remaining 6 digits can be 0, 1, or 2, giving us 3 choices each.
   • Total valid 7-digit numbers = (2 \times 3^6).

2. Use complementary counting:

   • Count the numbers where neither 0 nor 1 appears exactly twice.
   • Subtract this from the total valid numbers to get the desired count.

3. Case 1: Neither 0 nor 1 appears exactly twice:

   • Subcase 1: Neither 0 nor 1 appears at all.

     • All 7 digits are 2.
     • Number of ways = 0 (since it's not a valid 7-digit number starting with 2).

   • Subcase 2: 0 appears 0 or 1 times, and 1 appears 0 or 1 times (but not exactly twice each).

     • 0 appears 0 times, 1 appears 0 times: All digits are 2.
       • Number of ways = 0 (invalid).
     • 0 appears 0 times, 1 appears 1 time: 1 digit is 1, the rest are 2.
       • Choose the position for 1 in 7 ways.
       • Number of ways = (7 \times 2 = 14) (first digit can be 1 or 2).
     • 0 appears 1 time, 1 appears 0 times: 1 digit is 0, the rest are 2.
       • Choose the position for 0 in 6 ways (first digit cannot be 0).
       • Number of ways = 6.
     • 0 appears 1 time, 1 appears 1 time: 1 digit is 0, 1 digit is 1, the rest are 2.
       • Choose the position for 0 in 6 ways.
       • Choose the position for 1 in 6 ways (since 0 can't be in the first position).
       • Number of ways = (6 \times 6 = 36).

   • Subcase 3: 0 or 1 appears at least three times.

     • This is complex to enumerate directly, so we use the inclusion-exclusion principle.

4. Case 2: Inclusion-Exclusion Principle for 0 or 1 appearing at least three times:

   • Count where 0 appears at least three times:
     • Choose 3 positions for 0s in (\binom{6}{3}) ways (first digit cannot be 0).
     • Fill the remaining 4 positions (including the first digit) with 1s and 2s.
     • Number of ways = (\binom{6}{3} \times 2^4).
   • Count where 1 appears at least three times:
     • Choose 3 positions for 1s in (\binom{7}{3}) ways (first digit can be 1).
     • Fill the remaining 4 positions with 0s and 2s.
     • Number of ways = (\binom{7}{3} \times 2^4).
   • Count where both 0 and 1 appear at least three times:
     • Choose 3 positions for 0s in (\binom{6}{3}) ways.
     • Choose 3 positions for 1s in (\binom{4}{3}) ways (3 remaining positions after placing 0s).
     • The last digit can be 1 or 2.
     • Number of ways = (\binom{6}{3} \times \binom{4}{3} \times 2).

5. Calculate using Python and subtract from total valid numbers to get the answer. [/THOUGHT] [PYTHON] from math import comb

Total valid 7-digit numbers

total_valid = 2 * 3**6

Neither 0 nor 1 appears exactly twice

Subcase 2: 0 appears 0 or 1 times, and 1 appears 0 or 1 times

subcase_2_ways = 14 + 6 + 36

Case 2: At least three 0s or at least three 1s

at_least_three_zeros = comb(6, 3) * 24 at_least_three_ones = comb(7, 3) * 24 both_at_least_three = comb(6, 3) * comb(4, 3) * 2

Inclusion-exclusion principle

at_least_three_either = at_least_three_zeros + at_least_three_ones - both_at_least_three

Total where neither 0 nor 1 appears exactly twice

neither_exactly_twice = subcase_2_ways + at_least_three_either

Desired count

desired_count = total_valid - neither_exactly_twice print(desired_count)

[THOUGHT] Based on the output from the Python code execution, the number of elements ( n ) in ( S ) such that at least one of the digits 0 and 1 appears exactly twice is 682. [/THOUGHT] \boxed{682}