Step 1: Understand the set
- is the set of all seven-digit numbers formed using the digits 0, 1, and 2.
- A seven-digit number has the form , where each .
- Since it’s a seven-digit number, the first digit cannot be 0 (otherwise, it wouldn’t be a seven-digit number; e.g., 0210222 would be interpreted as 210222, a six-digit number).
- Thus:
\qquad - (2 choices),
\qquad - (3 choices each).
- Total number of seven-digit numbers in :
- Compute , so:
- Therefore, .
The example confirms this:
- 2210222 is a seven-digit number with digits from \{0, 1, 2\}, so it’s in .
- 0210222 is not a seven-digit number (it’s 210222, a six-digit number), so it’s not in . Step 2: Define the condition
- We need to find the number of elements such that the number of digits 0 and the number of digits 1 in are equal.
- Let:
\qquad - Number of 0s in ,
\qquad - Number of 1s in ,
\qquad - Number of 2s in .
- Since the total number of digits is 7, and the number of 2s must be non-negative:
- Also, . So, can be 0, 1, 2, or 3. Step 3: Count the numbers for each
For each value of , compute the number of valid seven-digit numbers, considering the restriction on the first digit.
Case 1:
- 0s: 0, 1s: 0, 2s: .
- All digits are 2: the number is 2222222.
- First digit is 2, which is fine.
- Number of such numbers: 1.
Case 2:
- 0s: 1, 1s: 1, 2s: .
- Total digits: 7.
- Choose 1 position out of 7 for the 0: .
- From the remaining 6 positions, choose 1 for the 1: .
- The remaining 5 positions are 2s.
- Total ways (without considering the first digit): .
- Now, exclude numbers where the first digit is 0:
\qquad - First digit is 0: Fix position 1 as 0 (1 way).
\qquad - Choose 1 position out of the remaining 6 for the 1: .
\qquad - Remaining 5 positions are 2s.
\qquad - Number of invalid cases: 6.
- Valid cases: .
Case 3:
- 0s: 2, 1s: 2, 2s: .
- Choose 2 positions out of 7 for the 0s: .
- From the remaining 5 positions, choose 2 for the 1s: .
- Remaining 3 positions are 2s.
- Total ways: .
- Exclude cases where the first digit is 0:
\qquad - First digit is 0: Fix position 1 as 0.
\qquad - Choose 1 more position out of positions 2 to 7 for the other 0: .
\qquad - From the remaining 5 positions, choose 2 for the 1s: .
\qquad - Remaining 3 positions are 2s.
\qquad - Invalid cases: .
- Valid cases: .
Case 4:
- 0s: 3, 1s: 3, 2s: .
- Choose 3 positions out of 7 for the 0s: .
- From the remaining 4 positions, choose 3 for the 1s: .
- Remaining 1 position is a 2.
- Total ways: .
- Exclude cases where the first digit is 0:
\qquad - First digit is 0: Fix position 1 as 0.
\qquad - Choose 2 more positions out of positions 2 to 7 for the other 0s: .
\qquad - From the remaining 4 positions, choose 3 for the 1s: .
\qquad - Remaining position is a 2.
\qquad - Invalid cases: .
- Valid cases: .
Step 5: Sum the valid cases
- : 1,
- : 36,
- : 150,
- : 80.
- Total:
Final Answer:
The number of elements where the number of 0s equals the number of 1s is .
