------
## 最终结论
```python
def countPermutations(S: str) -> int:
    MOD = 10**9 + 7
    n = len(S)
    
    # dp[i][j] represents the number of valid permutations of the first i characters ending with j
    dp = [[0] * (n + 1) for _ in range(n + 1)]
    
    # Base case: there's one way to arrange an empty string
    for j in range(n + 1):
        dp[0][j] = 1
    
    # Fill the dp table
    for i in range(1, n + 1):
        if S[i - 1] == 'D':
            # For 'D', we need to sum all dp[i-1][k] where k > j
            for j in range(n + 1):
                dp[i][j] = sum(dp[i - 1][k] for k in range(j + 1, n + 1)) % MOD
        else:  # S[i - 1] == 'I'
            # For 'I', we need to sum all dp[i-1][k] where k < j
            for j in range(n + 1):
                dp[i][j] = sum(dp[i - 1][k] for k in range(j)) % MOD
    
    # The result is the sum of all valid permutations of length n
    return sum(dp[n]) % MOD

# Example usage:
S = "DID"
print(countPermutations(S))  # Output: 5
```