What is the first term in the expansion of $(2 x+10 y)^{15}$?
To find the first term of the binomial expansion of \((2x + 10y)^{15}\), we use the Binomial Theorem. The Binomial Theorem states that for any non-negative integer \(n\), the expansion of \((A + B)^n\) is given by:
\[(A + B)^n = \sum_{k=0}^{n} \binom{n}{k} A^{n-k} B^k = \binom{n}{0} A^n B^0 + \binom{n}{1} A^{n-1} B^1 + \dots + \binom{n}{n} A^0 B^n\]
In our specific problem, we have \(A = 2x\), \(B = 10y\), and \(n = 15\). We want to find the first term, which corresponds to the case \(k=0\) in the summation.
The first term of the expansion corresponds to \(k=0\), given by the formula \(\binom{n}{0} A^n B^0\). Substituting our values \(A = 2x\), \(B = 10y\), and \(n = 15\) into this expression, we get:
\[\binom{15}{0} (2x)^{15} (10y)^0\]
Since \(\binom{15}{0} = 1\) and any non-zero term to the power of 0 is 1 (i.e., \((10y)^0 = 1\)), the expression simplifies significantly:
\[1 \cdot (2x)^{15} \cdot 1 = (2x)^{15}\]
Applying the power rule \((ab)^n = a^n b^n\), we have \((2x)^{15} = 2^{15} x^{15}\).
Comparing our result \((2^{15}x^{15})\) with the provided options:
We can clearly see that the correct choice is c. Thus, the first term in the expansion is \(\boxed{2^{15}x^{15}}\).
The binomial theorem is widely used in probability and statistics, especially when calculating the probabilities of events with two possible outcomes (like flipping a coin or passing/failing a test). For example, if you want to determine the probability of getting exactly 10 heads in 15 coin flips, the term \(\binom{15}{10} (0.5)^{10} (0.5)^5\) from the expansion of \((0.5+0.5)^{15}\) gives the solution. Understanding how to find individual terms helps in analyzing patterns and trends in data across many fields, from physics to finance.