The greatest coefficient in the expansion of ` (1 + x)^(10)`, is

Options

A. `(10!)/(5!6!)`
B. `(10!)/((5!)^(2))`
C. `(10!)/((5!7!))`
D. none of these

Answer

To find the greatest coefficient in the expansion of \( (1 + x)^{10} \), we can follow these steps:

### Step 1: Identify the general term in the binomial expansion
The general term \( T_r \) in the binomial expansion of \( (1 + x)^n \) is given by:
\[ T_r = \binom{n}{r} x^r \]
where \( n = 10 \) in our case. Thus, the general term becomes:
\[ T_r = \binom{10}{r} x^r \]

### Step 2: Find the condition for the greatest coefficient
To find the greatest coefficient, we need to determine the value of \( r \) for which \( T_r \) is maximized. The coefficients are maximized when:
\[ \frac{T_r}{T_{r-1}} \geq 1 \]
This leads us to:
\[ \frac{\binom{10}{r}}{\binom{10}{r-1}} \geq 1 \]
Using the property of binomial coefficients, we can simplify this to:
\[ \frac{10 - r + 1}{r} \geq 1 \]

### Step 3: Solve the inequality
This simplifies to:
\[ 11 - r \geq r \]
\[ 11 \geq 2r \]
\[ r \leq 5.5 \]
Since \( r \) must be an integer, the maximum integer value for \( r \) is \( 5 \).

### Step 4: Calculate the greatest coefficient
Now, we substitute \( r = 5 \) into the general term to find the coefficient:
\[ T_5 = \binom{10}{5} x^5 \]
The coefficient is:
\[ \binom{10}{5} = \frac{10!}{5!5!} \]

### Step 5: Calculate \( \binom{10}{5} \)
Calculating \( \binom{10}{5} \):
\[ \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = \frac{30240}{120} = 252 \]

### Conclusion
Thus, the greatest coefficient in the expansion of \( (1 + x)^{10} \) is \( 252 \).

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