To solve the problem of finding the 97th five-digit number where each successive digit exceeds its predecessor, we can follow these steps:
### Step 1: Understand the Problem
We need to find five-digit numbers formed by digits from 1 to 9, where each digit is greater than the previous one. This means we can only use combinations of digits without repetition.
### Step 2: Calculate Total Combinations
The total number of ways to choose 5 digits from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} is given by the combination formula \( C(n, r) \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
\[
C(9, 5) = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
\]
### Step 3: List the Combinations
Since we need to find the 97th combination, we can start listing the combinations in increasing order.
1. **Starting with the smallest digit (1)**:
- The combinations starting with 1 are formed by choosing 4 more digits from {2, 3, 4, 5, 6, 7, 8, 9}.
- The number of such combinations is \( C(8, 4) = 70 \).
2. **Next, starting with 2**:
- The combinations starting with 2 are formed by choosing 4 more digits from {3, 4, 5, 6, 7, 8, 9}.
- The number of such combinations is \( C(7, 4) = 35 \).
### Step 4: Count the Combinations
- Combinations starting with 1: 70
- Combinations starting with 2: 35
- Total combinations up to this point: \( 70 + 35 = 105 \)
Since 97 is less than 105, the 97th number must start with 2.
### Step 5: Fix the First Digit
Now that we know the first digit is 2, we need to find the 27th combination of the remaining digits (since \( 97 - 70 = 27 \)) from the set {3, 4, 5, 6, 7, 8, 9}.
### Step 6: Find the 27th Combination
1. **Starting with 2 and 3**:
- The combinations starting with 2 and 3 are formed by choosing 3 more digits from {4, 5, 6, 7, 8, 9}.
- The number of such combinations is \( C(6, 3) = 20 \).
2. **Next, starting with 2 and 4**:
- The combinations starting with 2 and 4 are formed by choosing 3 more digits from {5, 6, 7, 8, 9}.
- The number of such combinations is \( C(5, 3) = 10 \).
### Step 7: Count the Combinations Again
- Combinations starting with 2 and 3: 20
- Combinations starting with 2 and 4: 10
- Total combinations up to this point: \( 20 + 10 = 30 \)
Since 27 is less than 30, the 27th number must start with 2 and 4.
### Step 8: Fix the Second Digit
Now we need to find the 7th combination of the remaining digits from the set {5, 6, 7, 8, 9}.
### Step 9: Find the 7th Combination
1. **Starting with 2, 4, and 5**:
- The combinations starting with 2, 4, and 5 are formed by choosing 2 more digits from {6, 7, 8, 9}.
- The number of such combinations is \( C(4, 2) = 6 \).
2. **Next, starting with 2, 4, and 6**:
- The combinations starting with 2, 4, and 6 are formed by choosing 2 more digits from {7, 8, 9}.
- The number of such combinations is \( C(3, 2) = 3 \).
### Step 10: Count the Combinations Again
- Combinations starting with 2, 4, 5: 6
- Combinations starting with 2, 4, 6: 3
- Total combinations up to this point: \( 6 + 3 = 9 \)
Since 7 is less than 9, the 7th number must start with 2, 4, and 6.
### Step 11: Fix the Third Digit
Now we need to find the 1st combination of the remaining digits from the set {7, 8, 9}.
### Step 12: Find the 1st Combination
The first combination is simply {7, 8}.
### Final Answer
Thus, the 97th five-digit number in which each successive digit exceeds its predecessor is **24678**.
### Step 13: Identify the Missing Digit
Now, we check which digit is not present in 24678. The options given are 4, 5, 7, and 8. The digit that does not appear in the number is **5**.
