If `""^(100)C_(6)+4." "^(100)C_(7)+6." "^(100)C_(8)+4." "^(100)C_(9)+""^(100)C_(10)` has the value equal to `" "^(x)C_(y)` , then the possible value (s) of `x+y` can be :

Options

A. 112
B. 114
C. 196
D. 198

Answer

To solve the problem, we need to evaluate the expression:

\[ ^{100}C_6 + 4 \cdot ^{100}C_7 + 6 \cdot ^{100}C_8 + 4 \cdot ^{100}C_9 + ^{100}C_{10} \]

and express it in the form \(^{x}C_{y}\).

### Step 1: Group the terms
We can group the terms in a way that allows us to use the binomial coefficient identity:

\[ ^{n}C_{r} + ^{n}C_{r+1} = ^{n+1}C_{r+1} \]

We start by rewriting the expression:

\[ ^{100}C_6 + 4 \cdot ^{100}C_7 + 6 \cdot ^{100}C_8 + 4 \cdot ^{100}C_9 + ^{100}C_{10} \]

This can be rearranged as:

\[ ^{100}C_6 + (4 \cdot ^{100}C_7) + (6 \cdot ^{100}C_8) + (4 \cdot ^{100}C_9) + ^{100}C_{10} \]

### Step 2: Use the binomial coefficient identity
We can express \(4 \cdot ^{100}C_7\) as:

\[ ^{100}C_7 + ^{100}C_7 + ^{100}C_7 + ^{100}C_7 = 4 \cdot ^{100}C_7 \]

This allows us to combine terms:

\[ ^{100}C_6 + ^{100}C_7 + ^{100}C_7 + ^{100}C_7 + ^{100}C_7 + 6 \cdot ^{100}C_8 + 4 \cdot ^{100}C_9 + ^{100}C_{10} \]

### Step 3: Combine coefficients
Now we can combine the coefficients of \(^{100}C_7\) and \(^{100}C_8\):

\[ ^{100}C_6 + 4 \cdot ^{100}C_7 + 6 \cdot ^{100}C_8 + 4 \cdot ^{100}C_9 + ^{100}C_{10} \]

This can be rewritten as:

\[ ^{100}C_6 + ^{100}C_7 + ^{100}C_7 + ^{100}C_7 + ^{100}C_7 + ^{100}C_8 + ^{100}C_8 + ^{100}C_8 + ^{100}C_8 + ^{100}C_8 + ^{100}C_9 + ^{100}C_9 + ^{100}C_{10} \]

### Step 4: Use the identity repeatedly
Now we can use the binomial coefficient identity repeatedly:

1. Combine \(^{100}C_6\) and \(^{100}C_7\):
\[ ^{100}C_6 + ^{100}C_7 = ^{101}C_7 \]

2. Combine the next terms:
\[ ^{101}C_7 + ^{100}C_8 + ^{100}C_8 + ^{100}C_8 + ^{100}C_8 + ^{100}C_8 = ^{101}C_8 \]

3. Continue this process until we combine all terms.

### Step 5: Final combination
After combining all terms, we find that:

\[ ^{100}C_6 + 4 \cdot ^{100}C_7 + 6 \cdot ^{100}C_8 + 4 \cdot ^{100}C_9 + ^{100}C_{10} = ^{104}C_{10} \]

### Step 6: Find \(x + y\)
Here, \(x = 104\) and \(y = 10\). Thus:

\[ x + y = 104 + 10 = 114 \]

### Final Answer
The possible value(s) of \(x + y\) can be:

\[ \boxed{114} \]