Find the value of `(.^(10)C_(10))+(.^(10)C_(0)+.^(10)C_(1))+(.^(10)C_(0)+.^(10)C_(1)+.^(10)C_(2))+"...."+(.^(10)C_(0)+.^(10)C_(1)+.^(10)C_(2)+"....." + .^(10)C_(9))`.

Explanation

To solve the problem, we need to calculate the sum of several combinations. Let's denote the expression we want to evaluate as \( S \):

\[ S = \binom{10}{10} + \left( \binom{10}{0} + \binom{10}{1} \right) + \left( \binom{10}{0} + \binom{10}{1} + \binom{10}{2} \right) + \ldots + \left( \binom{10}{0} + \binom{10}{1} + \ldots + \binom{10}{9} \right) \]

### Step 1: Rewrite the expression

We can rewrite \( S \) more clearly by breaking it down:

\[ S = \binom{10}{10} + \sum_{k=0}^{9} \sum_{j=0}^{k} \binom{10}{j} \]

### Step 2: Simplify the inner sum

The inner sum \( \sum_{j=0}^{k} \binom{10}{j} \) represents the sum of the first \( k+1 \) terms of the binomial coefficients from \( \binom{10}{0} \) to \( \binom{10}{k} \).

### Step 3: Use the Binomial Theorem

According to the Binomial Theorem, we know that:

\[ \sum_{j=0}^{n} \binom{n}{j} = 2^n \]

Thus, for \( n = 10 \):

\[ \sum_{j=0}^{10} \binom{10}{j} = 2^{10} \]

### Step 4: Calculate the total contributions

The total contributions can be calculated as follows:

- The first term \( \binom{10}{10} = 1 \).
- The second term \( \sum_{j=0}^{0} \binom{10}{j} = \binom{10}{0} = 1 \).
- The third term \( \sum_{j=0}^{1} \binom{10}{j} = \binom{10}{0} + \binom{10}{1} = 1 + 10 = 11 \).
- The fourth term \( \sum_{j=0}^{2} \binom{10}{j} = \binom{10}{0} + \binom{10}{1} + \binom{10}{2} = 1 + 10 + 45 = 56 \).
- Continuing this way, we can see that the last term will be \( \sum_{j=0}^{9} \binom{10}{j} = 2^{10} - \binom{10}{10} = 1024 - 1 = 1023 \).

### Step 5: Combine all contributions

Now, we can express \( S \) as:

\[ S = 1 + 1 + 11 + 56 + \ldots + 1023 \]

### Step 6: Recognize the pattern

Notice that the inner sums can be expressed as:

\[ S = \sum_{k=0}^{10} \sum_{j=0}^{k} \binom{10}{j} = \sum_{k=0}^{9} (2^{10} - \binom{10}{10}) = 10 \cdot 2^{9} \]

### Final Calculation:

Thus, the final value of \( S \) is:

\[ S = 10 \cdot 2^{9} = 10 \cdot 512 = 5120 \]

### Conclusion

The value of \( S \) is \( 5120 \).

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