The value of -10r - 2 rC2 - 10Cr is not divisible by

∑^10_r = 2 ^rC_2· ^10C_r = ∑^10_r = 2r(r 1)2× ^10C_r nbsp;=∑^10_r = 2r^2 r2× ^10C_r =∑^10_r = 212× r^2 × ^10C_r 12∑^10_r = 2 r × ^10C_r (1 + x)^n = ^ ...

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Byju's Answer

Standard XI

Mathematics

Differentiation to Solve Modified Sum of Binomial Coefficients

The value of ...

Question

The value of $10 \sum r = 2^{r} C_{2} \cdot^{10} C_{r}$ is not divisible by

10

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5

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7

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3

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Solution

The correct option is C $7$
$10 \sum r = 2^{r} C_{2} \cdot^{10} C_{r} = 10 \sum r = 2 \frac{r (r - 1)}{2} \times^{10} C_{r}$
$= 10 \sum r = 2 \frac{r^{2} - r}{2} \times^{10} C_{r}$
$= 10 \sum r = 2 \frac{1}{2} \times r^{2} \times^{10} C_{r} - \frac{1}{2} 10 \sum r = 2 r \times^{10} C_{r}$

$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} + . . . +^{n} C_{n} x^{n}$
Differentiating both sides, we get
$n (1 + x)^{n - 1} =^{n} C_{1} + 2 \cdot^{n} C_{2} x + 3 \cdot^{n} C_{3} x^{2} + . . . + n \cdot^{n} C_{n} x^{n - 1} . . . (1)$
Let $x = 1$
$n \times 2^{n - 1} =^{n} C_{1} + 2 \cdot^{n} C_{2} + 3 \cdot^{n} C_{3} + 4 \cdot^{n} C_{4} + . . . + n \cdot^{n} C_{n}$
$= n \sum r = 1 r^{n} C_{r}$

Eqn. $(1) \times x$
$n x (1 + x)^{n - 1} = x \cdot^{n} C_{1} + 2 x^{2} \cdot^{n} C_{2} + 3 x^{3} \cdot^{n} C_{3} + . . . + n x^{n} \cdot^{n} C_{n}$
Differentiating
$n (1 + x)^{n - 1} + n (n - 1) x (1 + x)^{n - 2} =^{n} C_{1} + 2^{2}^{n} C_{2} x + 3^{2}^{n} C_{3} x^{2} + . . . + n^{2} x^{n - 1}^{n} C_{n}$
Put $x = 1$
$n \cdot 2^{n - 1} + n (n - 1) 2^{n - 2} = 1^{2}^{n} C_{1} + 2^{2}^{n} C_{2} + . . . + n^{2}^{n} C_{n}$
$\Rightarrow n \cdot 2^{n - 1} + n (n - 1) 2^{n - 2} = n \sum r = 1 r^{2}^{n} C_{r}$

$∴ 10 \sum r = 2 \frac{1}{2} r^{2}^{10} C_{r} - \frac{1}{2} 10 \sum r = 2 r \cdot^{10} C_{r}$
$= \frac{1}{2} [10 \cdot 2^{9} + 10 \cdot 9 \cdot 2^{8} - 10] - \frac{1}{2} [10 \cdot 2^{9} - 10]$
$= 10 \cdot 2^{8} + 10 \cdot 9 \cdot 2^{7} - 10 \cdot 2^{8}$
$= 10 \cdot 9 \cdot 2^{7}$
which is not divisible by $7$ .

Suggest Corrections

The value of 10∑r=2 rC2⋅ 10Cr is not divisible by

The value of $10 \sum r = 2^{r} C_{2} \cdot^{10} C_{r}$ is not divisible by