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

Answer: (D)

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:

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