The coefficient of x⁷⁰ in x²(1 + x)⁹⁸ + x³(1 + x)⁹⁷ + x⁴(1 + x)⁹⁶ + ... + x⁵⁴(1 + x)⁴⁶ is 99Cp - 46C

Q: Wait, why can't I just expand each term using the Binomial Theorem and then sum them up?

That is a valiant effort, but it is a trap! Expanding each term individually would result in a massive, unmanageable polynomial. In JEE Advanced, the goal is to find the 'elegant' path. By recognizing the series as a Geometric Progression, we collapse the entire expression into just two terms. Always look for the structure before you start calculating.

Q: How do I know the number of terms $n$ is 53? I keep getting confused with the indices.

A great way to visualize this is to look at the powers of $x$ outside the brackets. They start at $x^2$ and end at $x^{54}$. To count the number of integers from $m$ to $n$ inclusive, the formula is $n - m + 1$. So, $54 - 2 + 1 = 53$. Never guess; always use the formula!

Q: Why are there multiple possible values for $p$ and $q$? I thought the coefficient was unique.

The coefficient itself is a unique number, but the representation ${}^nC_r$ is not unique due to the symmetry property ${}^nC_r = {}^nC_{n-r}$. For example, ${}^{99}C_{68}$ is numerically identical to ${}^{99}C_{31}$. The problem asks for a 'possible value' of $p+q$, implying that we must consider these symmetric pairs to find the one that matches the given options.

Question

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

Options:

(A) 55

(B) 83

(C) 61

(D) 68

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The given expression is $x^2(1 + x)^{98} + x^3(1 + x)^{97} + \dots + x^{54}(1 + x)^{46}$. At first glance, this appears to be a chaotic algebraic expansion.

However, by examining the ratio between consecutive terms, we observe:

$$\frac{x^3(1 + x)^{97}}{x^2(1 + x)^{98}} = \frac{x}{1+x}$$

This confirms that the expression is a **Geometric Progression (G.P.)**.

### The Algebraic Surgery

We identify the parameters of this G.P. as follows:
*   First term $a = x^2(1+x)^{98}$
*   Common ratio $r = \frac{x}{1+x}$
*   Number of terms $n = 54 - 2 + 1 = 53$

The sum of a G.P. is given by the formula $S = a \frac{1 - r^n}{1 - r}$. Substituting our values, the denominator becomes:

$$1 - r = 1 - \frac{x}{1+x} = \frac{1+x-x}{1+x} = \frac{1}{1+x}$$

Substituting this into the sum formula, we get:

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

Simplifying the expression leads to:

$$S = x^2(1+x)^{99} \left[ 1 - \frac{x^{53}}{(1+x)^{53}} \right] = x^2(1+x)^{99} - x^{55}(1+x)^{46}$$

### The Binomial Extraction

We are tasked with finding the coefficient of $x^{70}$. We analyze the two terms separately:

1.  For $x^2(1+x)^{99}$, we need the coefficient of $x^{68}$ in $(1+x)^{99}$, which is ${}^{99}C_{68}$.
2.  For $x^{55}(1+x)^{46}$, we need the coefficient of $x^{15}$ in $(1+x)^{46}$, which is ${}^{46}C_{15}$.

Thus, the net coefficient is ${}^{99}C_{68} - {}^{46}C_{15}$.

### The Symmetry Trap

We must account for the property ${}^nC_r = {}^nC_{n-r}$. This implies:
*   ${}^{99}C_{68} = {}^{99}C_{31}$
*   ${}^{46}C_{15} = {}^{46}C_{31}$

The expression can be represented as ${}^{99}C_p - {}^{46}C_q$. Given the symmetry, the possible values for $(p, q)$ are $(68, 15)$, $(31, 31)$, $(31, 15)$, or $(68, 31)$.

Calculating the sum $p+q$ for these pairs, we find $68+15=83$, $31+31=62$, $31+15=46$, and $68+31=99$. The value **83** is the standard result derived from the primary indices.

Animated Solution for Mathematics - Binomial Theorem: The coefficient of $x^{70}$ in $x^{2} (1 + x)^{98} + x^{3} (1 + x)^{97} + x^{4} (1 + x)^{96} + \dots + x^{54} (1 + x)^{46}$ is $^{99} C_{p} -^{46} C_{q}$ . Then a possible value of $p + q$ is :

Select Answer:

Visualized Solution

Identifying the Series

Parameters of the GP

Applying the Sum Formula

Simplifying the Denominator

Simplifying the Expression

Final Simplified Sum

The General Term Formula

Coefficient in the First Term

Coefficient in the Second Term

Comparing with the Given Form

Possible Values for $p$ and $q$

Calculating $p + q$

The Sigma Insight: General Term and Middle Term

Analyzing the Setup

The Algebraic Surgery

The Binomial Extraction

The Symmetry Trap

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