The number of 7 digit integers abcdefg, where `a lt b lt c lt d gt e gt f gt g` such that a, b, c, d, e, f, g in {1,2,3,…..,9}`. Are

Options

A. 700
B. 20
C. 720
D. 800

Answer

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 cases based on the value of \( d \).

### Step-by-Step Solution:

1. **Understanding the constraints**:
- We need to select 4 digits for \( a, b, c, d \) such that \( a < b < c < d \).
- We also need to select 3 digits for \( e, f, g \) such that \( e > f > g \).
- The digits must be chosen from the set \( \{1, 2, 3, \ldots, 9\} \).

2. **Case Analysis Based on \( d \)**:
- We will analyze the cases based on the value of \( d \) (which can take values from 4 to 9, since we need at least three smaller digits for \( a, b, c \) and three smaller digits for \( e, f, g \)).

3. **Case 1: \( d = 4 \)**:
- Possible digits for \( a, b, c \) are \( \{1, 2, 3\} \) (3 choices).
- Possible digits for \( e, f, g \) are \( \{5, 6, 7, 8, 9\} \) (5 choices).
- Choose 3 from 5: \( \binom{5}{3} = 10 \).
- Total for this case: \( \binom{3}{3} \times 10 = 1 \times 10 = 10 \).

4. **Case 2: \( d = 5 \)**:
- Possible digits for \( a, b, c \) are \( \{1, 2, 3, 4\} \) (4 choices).
- Possible digits for \( e, f, g \) are \( \{6, 7, 8, 9\} \) (4 choices).
- Choose 3 from 4: \( \binom{4}{3} = 4 \).
- Total for this case: \( \binom{4}{3} \times 4 = 4 \times 4 = 16 \).

5. **Case 3: \( d = 6 \)**:
- Possible digits for \( a, b, c \) are \( \{1, 2, 3, 4, 5\} \) (5 choices).
- Possible digits for \( e, f, g \) are \( \{7, 8, 9\} \) (3 choices).
- Choose 3 from 3: \( \binom{3}{3} = 1 \).
- Total for this case: \( \binom{5}{3} \times 1 = 10 \times 1 = 10 \).

6. **Case 4: \( d = 7 \)**:
- Possible digits for \( a, b, c \) are \( \{1, 2, 3, 4, 5, 6\} \) (6 choices).
- Possible digits for \( e, f, g \) are \( \{8, 9\} \) (2 choices).
- Choose 3 from 2: \( \binom{2}{3} = 0 \).
- Total for this case: \( 0 \).

7. **Case 5: \( d = 8 \)**:
- Possible digits for \( a, b, c \) are \( \{1, 2, 3, 4, 5, 6, 7\} \) (7 choices).
- Possible digits for \( e, f, g \) are \( \{9\} \) (1 choice).
- Choose 3 from 1: \( \binom{1}{3} = 0 \).
- Total for this case: \( 0 \).

8. **Case 6: \( d = 9 \)**:
- Possible digits for \( a, b, c \) are \( \{1, 2, 3, 4, 5, 6, 7, 8\} \) (8 choices).
- Possible digits for \( e, f, g \) are none (0 choices).
- Total for this case: \( 0 \).

### Final Calculation:
Now, we sum the totals from all the cases:
- Case 1: 10
- Case 2: 16
- Case 3: 10
- Case 4: 0
- Case 5: 0
- Case 6: 0

Total = \( 10 + 16 + 10 + 0 + 0 + 0 = 36 \).

### Final Answer:
The total number of 7-digit integers \( abcdefg \) such that \( a < b < c < d > e > f > g \) is **36**.