Find the sum of the series `.^(84)C_(4)+6xx.^(84)C_(5)+15xx.^(84)C_(6)+20xx.^(84)C_(7)+15xx.^(84)C_(8)``+6xx.^(84)C_(9)+.^(84)C_(10)`.

Explanation

To find the sum of the series \( \binom{84}{4} + 6 \cdot \binom{84}{5} + 15 \cdot \binom{84}{6} + 20 \cdot \binom{84}{7} + 15 \cdot \binom{84}{8} + 6 \cdot \binom{84}{9} + \binom{84}{10} \), we can use the Binomial Theorem and properties of binomial coefficients.

### Step-by-Step Solution:

1. **Rewrite the Series**:
We can express the coefficients in terms of binomial coefficients:
\[ \binom{84}{4} + 6 \cdot \binom{84}{5} + 15 \cdot \binom{84}{6} + 20 \cdot \binom{84}{7} + 15 \cdot \binom{84}{8} + 6 \cdot \binom{84}{9} + \binom{84}{10} \]
can be rewritten as:
\[ \binom{6}{6} \cdot \binom{84}{4} + \binom{6}{5} \cdot \binom{84}{5} + \binom{6}{4} \cdot \binom{84}{6} + \binom{6}{3} \cdot \binom{84}{7} + \binom{6}{2} \cdot \binom{84}{8} + \binom{6}{1} \cdot \binom{84}{9} + \binom{6}{0} \cdot \binom{84}{10} \]

2. **Identify the Generating Function**:
The sum can be interpreted as the coefficient of \( x^{10} \) in the expansion of:
\[ (1 + x^6)(1 + x^{84}) \]
This is because each term corresponds to choosing a certain number of \( x^6 \) from the first factor and \( x^{84} \) from the second factor.

3. **Expand the Generating Function**:
The generating function can be simplified:
\[ (1 + x^6)(1 + x^{84}) = 1 + x^6 + x^{84} + x^{90} \]

4. **Find the Coefficient of \( x^{10} \)**:
We need to find the coefficient of \( x^{10} \) in this expression. Since the highest power is \( x^{90} \), and \( x^{10} \) is less than \( x^6 \), the only relevant term is \( 1 \). Thus, the coefficient of \( x^{10} \) in \( 1 + x^6 + x^{84} + x^{90} \) is \( 0 \).

5. **Use the Binomial Coefficient**:
We can also express the sum as:
\[ \binom{90}{10} \]
This is derived from the fact that the coefficient of \( x^{10} \) in \( (1 + x)^{90} \) is \( \binom{90}{10} \).

### Final Result:
Thus, the sum of the series is:
\[ \boxed{\binom{90}{10}} \]