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1

### IIT-JEE 2010 Paper 2 Offline

For $$r = 0,\,1,....,$$ let $${A_r},\,{B_r}$$ and $${C_r}$$ denote, respectively, the coefficient of $${X^r}$$ in the expansions of $${\left( {1 + x} \right)^{10}},$$ $${\left( {1 + x} \right)^{20}}$$ and $${\left( {1 + x} \right)^{30}}.$$
Then $$\sum\limits_{r = 1}^{10} {{A_r}\left( {{B_{10}}{B_r} - {C_{10}}{A_r}} \right)}$$ is equal to
A
$$\left( {{B_{10}} - {C_{10}}} \right)$$
B
$${A_{10}}\left( {{B^2}_{10}{C_{10}}{A_{10}}} \right)$$
C
$$0$$
D
$${{C_{10}} - {B_{10}}}$$
2

### IIT-JEE 2005 Screening

The value of $$\left( {\matrix{ {30} \cr 0 \cr } } \right)\left( {\matrix{ {30} \cr {10} \cr } } \right) - \left( {\matrix{ {30} \cr 1 \cr } } \right)\left( {\matrix{ {30} \cr {11} \cr } } \right) + \left( {\matrix{ {30} \cr 2 \cr } } \right)\left( {\matrix{ {30} \cr {12} \cr } } \right)....... + \left( {\matrix{ {30} \cr {20} \cr } } \right)\left( {\matrix{ {30} \cr {30} \cr } } \right)$$\$
is where $$\left( {\matrix{ n \cr r \cr } } \right) = {}^n{C_r}$$
A
$$\left( {\matrix{ {30} \cr {10} \cr } } \right)$$
B
$$\left( {\matrix{ {30} \cr {15} \cr } } \right)$$
C
$$\left( {\matrix{ {60} \cr {30} \cr } } \right)$$
D
$$\left( {\matrix{ {31} \cr {10} \cr } } \right)$$
3

### IIT-JEE 2004 Screening

If $${}^{n - 1}{C_r} = \left( {{k^2} - 3} \right)\,{}^n{C_{r + 1,}}$$ then $$k \in$$
A
$$\left( { - \infty , - 2} \right)$$
B
$$\left[ {2,\infty } \right)$$
C
$$\left[ { - \sqrt 3 ,\sqrt 3 } \right]$$
D
$$\left( {\sqrt 3 ,2} \right]$$
4

### IIT-JEE 2003 Screening

Coefficient of $${t^{24}}$$ in $${\left( {1 + {t^2}} \right)^{12}}\left( {1 + {t^{12}}} \right)\left( {1 + {t^{24}}} \right)$$ is
A
$${}^{12}{C_6} + 3$$
B
$${}^{12}{C_6} + 1$$
C
$${}^{12}{C_6}$$
D
$${}^{12}{C_6} + 2$$

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