Let be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in , but 0210222 is NOT in .Find the number of elements in such that at least one of the digits 0 and 1 appears exactly twice in .\
[THOUGHT] To solve this, compute |A ∪ B| where A is numbers with exactly two 0's, B with exactly two 1's, both in S (leading digit 1 or 2).
Define a function to count valid arrangements for given counts of 0's (a), 1's (b), 2's (c):
Total arrangements: 7! / (a! b! c!)
Leading zero cases (if a >=1): 6! / ((a-1)! b! c!)
Valid = total - leading zero.
Then |A| = sum_{b=0 to 5} count_valid(2, b, 5-b)
|B| = sum_{a=0 to 5} count_valid(a, 2, 5-a)
|A ∩ B| = count_valid(2,2,3)
Result = |A| + |B| - |A ∩ B| [/THOUGHT] [PYTHON] import math
def multinomial(n, counts): res = math.factorial(n) for k in counts: res //= math.factorial(k) return res
def count_valid(a, b, c): if a + b + c != 7: return 0 total = multinomial(7, [a, b, c]) if a == 0: leading0 = 0 else: leading0 = multinomial(6, [a-1, b, c]) return total - leading0
sum_A = 0 for b in range(6): c = 5 - b sum_A += count_valid(2, b, c)
sum_B = 0 for a in range(6): c = 5 - a sum_B += count_valid(a, 2, c)
inter = count_valid(2, 2, 3)
result = sum_A + sum_B - inter print(result) [/PYTHON]
[THOUGHT] The Python code executed successfully and output 762, which is the exact count. [/THOUGHT] \boxed{762}
Explanation
The reasoning uses the inclusion-exclusion principle to count seven-digit numbers in S (no leading zero) where at least one of digits 0 or 1 appears exactly twice. Define set A as numbers with exactly two 0's and set B as numbers with exactly two 1's. The goal is |A ∪ B| = |A| + |B| - |A ∩ B|.
For given counts of 0's (a), 1's (b), and 2's (c) where a + b + c = 7, the total arrangements are the multinomial coefficient 7! / (a! b! c!). Subtract invalid cases with a leading zero: if a ≥ 1, these are 6! / ((a-1)! b! c!).
Thus, |A| sums valid counts for a=2 and b from 0 to 5 (c=5-b): ∑ count_valid(2, b, 5-b).
|B| sums valid counts for b=2 and a from 0 to 5 (c=5-a): ∑ count_valid(a, 2, 5-a).
|A ∩ B| is count_valid(2, 2, 3).
The Python code implements this: defines multinomial and count_valid functions, computes the sums, and prints |A| + |B| - |A ∩ B|, yielding 762 as verified output.