The coefficient of x 70 in x 2 ( 1 + x ) 98 + x 3 ( 1 + x ) 97 + x 4 ( 1 + x ) 96 + … + x 54 ( 1 + x ) 46 is C p - C q . 46 99 Then a possible value of p + q is: [2024]

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Solution

Q.
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: [2024]

1 68

2 83

3 55

4 61

Ans.
(2)

We have, $x^{2} {(1 + x)}^{98} + x^{3} {(1 + x)}^{97} + \dots + x^{54} {(1 + x)}^{46}$

It is a G.P. with first term $= x^{2} {(1 + x)}^{98}$

and common ratio $= \frac{x}{1 + x}$

$∴$ Sum of these terms $= x^{2} {(1 + x)}^{98} (\frac{{(\frac{x}{1 + x})}^{53} - 1}{\frac{x}{1 + x} - 1})$

$= x^{2} {(1 + x)}^{98} ((1 + x) - x^{53} {(1 + x)}^{- 52})$

$= x^{2} \underset{coeff . of x^{68}}{\underset{⏝}{{(1 + x)}^{99}}}$ $- x^{55}$ $\underset{coeff . of x^{15}}{\underset{⏝}{{(1 + x)}^{46}}}$ [ $∵$ we need coefficient of $x^{70}$ ]

$= {}^{99}{C_{68}} - {}^{46}{C_{15}} = {}^{99}{C_{p}} - {}^{46}{C_{q}}$ ( $∵$ Given)

Hence, $p + q = 83$

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