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

The term \( 35 x^{3} y^{4} \) appears in the binomial expansion of \( (x+y)^{7} \). In the expansion, the general term can be expressed as \( T_k = \binom{n}{k} x^{n-k} y^{k} \). For \( n = 7 \), setting \( n-k = 3 \) (which gives \( k = 4 \)), we find the coefficient \( \binom{7}{4} \) which equals 35, yielding the exact term \( 35 x^{3} y^{4} \). Meanwhile, consider how this ties into the real-world! The binomial expansion isn't just abstract math—engineers and scientists use it everyday in physics for calculating probabilities, distributions, and even optimizing designs based on varying parameters. So the next time you expand a binomial, remember it's not just a number crunch; it's real-world problem-solving at its best!