Let q be a positive with `q le 50`.
If the sum `""^(98)C_(30)+2" "^(97)C_(30)+3." "^(96)C_(30)+ …… + 68." "^(31)C_(30)+69." "^(30)C_(30)=""^(100)C_(q)`
Find the sum of the digits of q.

Answer

To solve the problem, we need to evaluate the left-hand side expression and relate it to the right-hand side expression. Let's break it down step by step.

### Step 1: Understanding the Left-Hand Side Expression
The left-hand side of the equation is:
\[ \sum_{k=0}^{68} (69-k) \binom{98-k}{30} \]
This expands to:
\[ 69 \binom{98}{30} + 68 \binom{97}{30} + 67 \binom{96}{30} + \ldots + 1 \binom{30}{30} \]

### Step 2: Rewriting the Left-Hand Side
We can rewrite the left-hand side using properties of binomial coefficients. We notice that:
\[ \sum_{k=0}^{68} (69-k) \binom{98-k}{30} = \sum_{k=0}^{68} \binom{98-k}{30} + \sum_{k=0}^{68} (69-k) \binom{98-k}{30} \]
This can be simplified using the hockey-stick identity in combinatorics.

### Step 3: Applying the Hockey-Stick Identity
Using the hockey-stick identity:
\[ \sum_{j=r}^{n} \binom{j}{r} = \binom{n+1}{r+1} \]
we can express the sum of the binomial coefficients in a more manageable form.

### Step 4: Relating to the Right-Hand Side
The right-hand side of the equation is:
\[ \binom{100}{q} \]
From our previous steps, we can find that the left-hand side simplifies to:
\[ \binom{100}{32} \]
This means we have:
\[ \binom{100}{q} = \binom{100}{32} \]

### Step 5: Finding q
From the properties of binomial coefficients, we know that:
\[ \binom{n}{k} = \binom{n}{n-k} \]
Thus, we have:
\[ q = 32 \quad \text{or} \quad q = 100 - 32 = 68 \]
Since \( q \) must be less than or equal to 50, we take:
\[ q = 32 \]

### Step 6: Finding the Sum of the Digits of q
Now, we need to find the sum of the digits of \( q \):
\[ q = 32 \implies 3 + 2 = 5 \]

### Final Answer
The sum of the digits of \( q \) is:
\[ \boxed{5} \]