What is the second term in the binomial expansion of $(2 r-3 s)^{12}$ ?
To find the second term in the binomial expansion of \((2r - 3s)^{12}\), we apply the Binomial Theorem. The theorem states that for any positive integer \(n\), the expansion is given by:
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
In this expression, we identify \(a = 2r\), \(b = -3s\), and \(n = 12\). The terms in the expansion are indexed by \(k\), starting from \(k = 0\). The first term corresponds to \(k = 0\), and the second term corresponds to \(k = 1\).
To find the second term, we substitute \(k = 1\) into the general formula \(\binom{n}{k} a^{n-k} b^k\):
\[\binom{12}{1} (2r)^{12-1} (-3s)^1\]
First, calculate the binomial coefficient:
\[\binom{12}{1} = \frac{12}{1} = 12\]
Next, evaluate the power of the first term \((2r)^{11}\):
\[(2r)^{11} = 2^{11} \cdot r^{11} = 2048 \cdot r^{11}\]
Finally, the expression for the second term is:
\[12 \cdot (2048 \cdot r^{11}) \cdot (-3s)\]
Now, multiply the coefficients and the variables:
\[12 \cdot 2048 \cdot (-3) = 24576 \cdot (-3) = -73728\]
Combining this with the variables \(r^{11}\) and \(s\), we get:
\[-73728 \cdot r^{11} \cdot s = -73728 r^{11} s\]
Comparing this to the given options, it matches option (d). Therefore, the second term is \(-73728 r^{11} s\).
The binomial theorem is widely used in probability and statistics, particularly when dealing with sequences of independent experiments like flipping a coin or calculating the likelihood of specific outcomes. For example, if you wanted to quickly determine the probability of getting exactly 2 heads in 12 coin tosses, you could use the binomial expansion to identify the relevant term without having to list every possible outcome combination. This technique is also fundamental in fields like physics and engineering, where complex polynomial expressions are approximated using these expansions.