To solve the problem of finding the number of 7-digit integers \( abcdefg \) such that \( a < b < c < d > e > f > g \) with \( a, b, c, d, e, f, g \in \{1, 2, 3, \ldots, 9\} \), we can break it down into steps.
### Step 1: Identify the constraints
We have the following constraints:
- \( a < b < c < d \)
- \( d > e > f > g \)
This means that \( a, b, c \) must be chosen from the numbers less than \( d \), and \( e, f, g \) must be chosen from the numbers less than \( d \) but greater than \( d \).
### Step 2: Choose values for \( d \)
The value of \( d \) can be any number from 4 to 9 (since we need at least three numbers less than \( d \) for \( a, b, c \) and three numbers greater than \( d \) for \( e, f, g \)).
### Step 3: Calculate combinations for each case of \( d \)
#### Case 1: \( d = 4 \)
- Possible values for \( a, b, c \) are from \( \{1, 2, 3\} \).
- We can choose 3 numbers from these 3: \( \binom{3}{3} = 1 \).
- No numbers are available for \( e, f, g \) since there are no numbers greater than 4.
- Total ways = \( 1 \times 0 = 0 \).
#### Case 2: \( d = 5 \)
- Possible values for \( a, b, c \) are from \( \{1, 2, 3, 4\} \).
- Choose 3 from these 4: \( \binom{4}{3} = 4 \).
- Possible values for \( e, f, g \) are from \( \{6, 7, 8, 9\} \).
- Choose 3 from these 4: \( \binom{4}{3} = 4 \).
- Total ways = \( 4 \times 4 = 16 \).
#### Case 3: \( d = 6 \)
- Possible values for \( a, b, c \) are from \( \{1, 2, 3, 4, 5\} \).
- Choose 3 from these 5: \( \binom{5}{3} = 10 \).
- Possible values for \( e, f, g \) are from \( \{7, 8, 9\} \).
- Choose 3 from these 3: \( \binom{3}{3} = 1 \).
- Total ways = \( 10 \times 1 = 10 \).
#### Case 4: \( d = 7 \)
- Possible values for \( a, b, c \) are from \( \{1, 2, 3, 4, 5, 6\} \).
- Choose 3 from these 6: \( \binom{6}{3} = 20 \).
- Possible values for \( e, f, g \) are from \( \{8, 9\} \).
- Choose 3 from these 2: \( \binom{2}{3} = 0 \).
- Total ways = \( 20 \times 0 = 0 \).
#### Case 5: \( d = 8 \)
- Possible values for \( a, b, c \) are from \( \{1, 2, 3, 4, 5, 6, 7\} \).
- Choose 3 from these 7: \( \binom{7}{3} = 35 \).
- Possible values for \( e, f, g \) are from \( \{9\} \).
- Choose 3 from these 1: \( \binom{1}{3} = 0 \).
- Total ways = \( 35 \times 0 = 0 \).
#### Case 6: \( d = 9 \)
- Possible values for \( a, b, c \) are from \( \{1, 2, 3, 4, 5, 6, 7, 8\} \).
- Choose 3 from these 8: \( \binom{8}{3} = 56 \).
- No numbers are available for \( e, f, g \) since there are no numbers greater than 9.
- Total ways = \( 56 \times 0 = 0 \).
### Step 4: Sum the total ways
Now we sum the total ways from each case:
- Total = \( 0 + 16 + 10 + 0 + 0 + 0 = 26 \).
### Final Answer
The total number of 7-digit integers \( abcdefg \) satisfying the given conditions is **26**.
