Question

If the coefficients of the first, second, and third terms in the expansion of \( (1+x)^n \) are in the ratio \(1:20:190\), then \(n\) is equal to:

• A. \(18\)
• B. \(19\)
• C. \(20\)
• D. \(21\)

Hint:

For the expansion of \((1+x)^n\), always remember the first few coefficients: \[ 1,\quad n,\quad \frac{n(n-1)}{2},\quad \frac{n(n-1)(n-2)}{6},\ldots \] Many competitive examination questions involving ratios of coefficients can be solved directly by comparing these standard expressions without performing a complete expansion.

Answer 1

The Correct Option is C

Step 1: Identify the coefficients.
In $(1+x)^n$, the first, second, and third term coefficients are $\binom{n}{0}=1$, $\binom{n}{1}=n$, and $\binom{n}{2}=\frac{n(n-1)}{2}$.

Step 2: Write the given ratio.
They are in the ratio $1:20:190$. So $1:n:\frac{n(n-1)}{2}=1:20:190$.

Step 3: Use the first two parts.
Comparing the first two: $\frac{n}{1}=\frac{20}{1}$, so $n=20$.

Step 4: Check with the next ratio.
Compare the second and third parts: $\frac{\binom{n}{2}}{\binom{n}{1}}=\frac{190}{20}$. The left side simplifies: $\frac{n(n-1)/2}{n}=\frac{n-1}{2}$.

Step 5: Solve the check.
So $\frac{n-1}{2}=\frac{190}{20}=9.5$. That gives $n-1=19$, so $n=20$. Both conditions agree.

Step 6: Confirm consistency.
With $n=20$: coefficients are $1$, $20$, and $\frac{20\times 19}{2}=190$. The ratio is exactly $1:20:190$, matching perfectly.

Step 7: State the answer.
\[ \boxed{20} \]