The greatest coefficient in the expansion of `(1+x)^(2n)` is

Options

A. `""^(2n)C_n`
B. `""^(2n)C_(n-1)`
C. `""^(2n)C_(n+1)`
D. None of the above

Answer

To find the greatest coefficient in the expansion of \((1+x)^{2n}\), we can follow these steps:

### Step 1: Identify the general term in the binomial expansion
The general term in the expansion of \((1+x)^{2n}\) is given by:
\[ T_r = \binom{2n}{r} x^r \]
where \(T_r\) is the \(r\)-th term, and \(\binom{2n}{r}\) is the binomial coefficient.

### Step 2: Determine the value of \(r\) that maximizes the coefficient
To find the greatest coefficient, we need to maximize \(\binom{2n}{r}\). The binomial coefficient \(\binom{2n}{r}\) reaches its maximum value when \(r\) is approximately equal to \(\frac{2n}{2} = n\).

### Step 3: Evaluate the maximum coefficient
Since \(2n\) is even, the maximum coefficient occurs at \(r = n\). Thus, the greatest coefficient is:
\[ \binom{2n}{n} \]

### Conclusion
The greatest coefficient in the expansion of \((1+x)^{2n}\) is:
\[ \boxed{\binom{2n}{n}} \]

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