Question

The coefficient of \( x^{70} \) in \( x^2 (1+x)^{98} + x^3 (1+x)^{97} + x^4 (1+x)^{96} + \ldots + x^{54} (1+x)^{46} \) is \( ^{99}C_p - ^{46}C_q \).
Then a possible value to \( p + q \) is:

• A. 55
• B. 61
• C. 68
• D. 83

Answer 1

The Correct Option is D

Given:

\( S = x^2(1 + x)^{98} + x^3(1 + x)^{97} + x^4(1 + x)^{96} + \ldots + x^{54}(1 + x)^{46} \)

It is a geometric progression (G.P.).

Therefore,

\( S = x^2(1 + x)^{98} \left[ \frac{\left( \frac{x}{1 + x} \right)^{53} - 1}{\frac{x}{1 + x} - 1} \right] \)

Now, the coefficient of \( x^{70} \) in \( S \) will be:

\( S = x^2(1 + x)^{46} \left[ (1 + x)^{53} - x^{53} \right] \)

The coefficient of \( x^{70} \) in \( S \) is obtained from:

\( S = x^2(1 + x)^{99} - x^{55}(1 + x)^{46} \)

Hence, the required coefficient is:

\( S = {}^{99}C_{68} - {}^{46}C_{15} = {}^{99}C_{p} - {}^{46}C_{q} \)

From this,

\( p = 68, \quad q = 15 \)

Therefore,

\( p + q = 83 \)

Answer 2

\[ x^2(1+x)^{98} + x^3(1+x)^{97} + x^4(1+x)^{96} + …… + x^{54}(1+x)^{46} \]

The coefficient of \(x^{70}\) is:

\[ ^{98}C_{68} + ^{97}C_{67} + ^{96}C_{66} + \cdots \]

Simplify:

\[ ^{47}C_{17} + ^{46}C_{16} \]

Combine terms:

\[ {^{46}}C_{30} + {^{46}}C_{31} + ^{47}C_30 + \cdots \]

Using binomial expansion:

\[ {^{47}}C_{30} + \cdots = ^{99}C_p - ^{46}C_q \]

Possible values of \(p+q\):

\[ p+q = 62, 83, 99, 46 \]

Final Answer:

\[ p+q = 83 \quad \text{Option (4)}. \]