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

Concept: The Binomial Theorem provides a systematic way to expand expressions of the form \[ (a+b)^n. \] According to the Binomial Theorem, \[ (a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r. \] In the expansion of \((1+x)^n\), the coefficients of the terms are precisely the binomial coefficients \[ \binom{n}{0},\binom{n}{1},\binom{n}{2},\binom{n}{3},\ldots \] Whenever a question provides a ratio involving coefficients of consecutive terms, the most efficient approach is to first write the coefficients explicitly using the standard binomial coefficient formulas and then compare them with the given ratio. This allows us to determine the unknown value of \(n\) directly without expanding the entire expression.

Step 1: Write the first three coefficients of the expansion. For the expansion \[ (1+x)^n, \] the first term is \[ \binom{n}{0}. \] The second term is \[ \binom{n}{1}x. \] The third term is \[ \binom{n}{2}x^2. \] Therefore, the corresponding coefficients are \[ \binom{n}{0},\qquad \binom{n}{1},\qquad \binom{n}{2}. \]

Step 2: Replace the coefficients by their standard values. Using the formulas for binomial coefficients, \[ \binom{n}{0}=1, \] \[ \binom{n}{1}=n, \] and \[ \binom{n}{2}=\frac{n(n-1)}{2}. \] Hence the ratio of the first three coefficients becomes \[ 1:n:\frac{n(n-1)}{2}. \]

Step 3: Compare the obtained ratio with the given ratio. The question states that the coefficients are in the ratio \[ 1:20:190. \] Comparing \[ 1:n:\frac{n(n-1)}{2} \] with \[ 1:20:190, \] we immediately obtain \[ n=20. \]

Step 4: Verify the value using the third coefficient. To ensure that the value obtained is correct, substitute \[ n=20 \] into the third coefficient. \[ \binom{20}{2} = \frac{20(20-1)}{2}. \] \[ = \frac{20\times19}{2}. \] \[ = 10\times19. \] \[ = 190. \] This matches the third part of the given ratio exactly.

Step 5: Check the entire ratio. Substituting \(n=20\), \[ 1:n:\frac{n(n-1)}{2} = 1:20:190. \] This is exactly the ratio given in the question. Therefore, our value of \(n\) satisfies all the conditions provided.

Step 6: Final Conclusion. The required value of \(n\) is \[ \boxed{20}. \] Hence, the correct answer is \[ \boxed{\text{Option (C)}}. \]