The term $$ 35 x^{3} y^{4} $$ is a term in which binomial expansion? $$ \frac{(x+y)^{12}}{(7 x+5 y)^{7}} $$ $$ \frac{(x+y)^{7}}{(x+y)^{6}} $$

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To determine in which binomial expansion the term $$ 35 x^{3} y^{4} $$ appears, we will analyze both expressions given.

1. **For the first expression**: 
   $$
   \frac{(x+y)^{12}}{(7x+5y)^{7}}
   $$

We can start by expanding both the numerator and the denominator.

- The numerator $$ (x+y)^{12} $$ can be expanded using the binomial theorem:
     $$
     (x+y)^{12} = \sum_{k=0}^{12} \binom{12}{k} x^{12-k} y^{k}
     $$

- The denominator $$ (7x + 5y)^{7} $$ can also be expanded:
     $$
     (7x + 5y)^{7} = \sum_{j=0}^{7} \binom{7}{j} (7x)^{7-j} (5y)^{j} = \sum_{j=0}^{7} \binom{7}{j} 7^{7-j} 5^{j} x^{7-j} y^{j}
     $$

To find the term $$ 35 x^{3} y^{4} $$, we need to consider the combined effect of both expansions.

The term $$ x^{3} y^{4} $$ in $$ (x+y)^{12} $$ corresponds to $$ k=4 $$:
   $$
   \binom{12}{4} x^{3} y^{4} = \binom{12}{4} x^{3} y^{4}
   $$
   where $$ \binom{12}{4} = 495 $$.

The term $$ x^{3} y^{4} $$ in the denominator $$ (7x + 5y)^{7} $$ corresponds to $$ j=4 $$:
   $$
   \binom{7}{4} 7^{3} 5^{4} x^{3} y^{4} = \binom{7}{4} 7^{3} 5^{4} x^{3} y^{4}
   $$
   where $$ \binom{7}{4} = 35 $$, $$ 7^{3} = 343 $$, and $$ 5^{4} = 625 $$.

Thus, the coefficient of $$ x^{3} y^{4} $$ in the denominator is:
   $$
   35 \cdot 343 \cdot 625
   $$

The overall coefficient of $$ x^{3} y^{4} $$ in the entire expression is:
   $$
   \frac{495}{35 \cdot 343 \cdot 625}
   $$

Since we are looking for the term $$ 35 x^{3} y^{4} $$, we can see that this term does not appear in this expression.

2. **For the second expression**:
   $$
   \frac{(x+y)^{7}}{(x+y)^{6}} = (x+y)^{1} = x+y
   $$

The term $$ 35 x^{3} y^{4} $$ cannot appear in $$ x+y $$ since it only contains the terms $$ x $$ and $$ y $$.

In conclusion, the term $$ 35 x^{3} y^{4} $$ does not appear in either of the given binomial expansions.
The term $$ 35 x^{3} y^{4} $$ does not appear in either of the given binomial expansions.

The term is a term in which binomial
expansion?

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