The value of - 30- 0 - 30- 10 - 30- 1 - 30- 11 - 30- 2 - 30- 12 - 30- 20 - 30- 20 -

The correct option is D (1 x)^30=^30C_0x^0 ^30C_1x^1+^30C_2x^2+⋯ ( 1)^30^30 C_30x^30 .............(i) (1+x)^30=^30C_0x^30+^30 C_1x^29+^30 C_2x^28+⋯^30C_10x^0+⋯ ...

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Byju's Answer

Standard XII

Mathematics

nth Term of A.P

The value of ...

Question

The value of $(\begin{matrix} 300 \end{matrix}) (\begin{matrix} 3010 \end{matrix}) - (\begin{matrix} 301 \end{matrix}) (\begin{matrix} 3011 \end{matrix}) + (\begin{matrix} 302 \end{matrix}) (\begin{matrix} 3012 \end{matrix}) + \dots + (\begin{matrix} 3020 \end{matrix}) (\begin{matrix} 3020 \end{matrix})$

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Solution

The correct option is B
$(1 - x)^{30} =^{30} C_{0} x^{0} -^{30} C_{1} x^{1} +^{30} C_{2} x^{2} + \dots (- 1)^{30}^{30} C_{30} x^{30}$ .............(i)
$(1 + x)^{30} =^{30} C_{0} x^{30} +^{30} C_{1} x^{29} +^{30} C_{2} x^{28} + \dots^{30} C_{10} x^{0} + \dots +^{30} C_{30} x^{0}$ .............. (ii)
Multiplying (i) and (ii) and equating the coefficient of $x^{20}$ on both sides, we get required sum = coefficient of $x^{20}$ in $(1 - x^{2})^{30} =^{30} C_{10}$

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Similar questions

Q. The value of

(\frac{30}{0}) (\frac{30}{10}) - (\frac{30}{1}) (\frac{30}{11}) + (\frac{30}{2}) (\frac{30}{12}) . . . + (\frac{30}{20}) (\frac{30}{30})

is; where

(\frac{n}{r}) =^{n} C_{r}

The value of $(\begin{matrix} 300 \end{matrix})$ $(\begin{matrix} 3010 \end{matrix})$ - $(\begin{matrix} 301 \end{matrix})$ $(\begin{matrix} 3011 \end{matrix})$

+ $(\begin{matrix} 302 \end{matrix})$ $(\begin{matrix} 3012 \end{matrix})$ + ..............+ $(\begin{matrix} 3020 \end{matrix})$ $(\begin{matrix} 3030 \end{matrix})$