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  • Find the value of <img alt="\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}+\cdots+\mathrm{C}_{n}\right)^{3}" src="http://entrancecorner.oncodecogs.com/gif.latex?%5Cleft%28%5Cmathrm%7BC%7D_

Find the value of \left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}+\cdots+\mathrm{C}_{n}\right)^{3} 

Option: 1

2^{3n} -1


Option: 2

1+\;^{3 n} \mathrm{C}_{1}+^{3 n} \mathrm{C}_{2}+\cdots+^{3 n} \mathrm{C}_{3 n}


Option: 3

2^{3n} +1


Option: 4

None of the above


Answers (1)

best_answer

Series Involving Binomial Coefficients

Sum of Coefficient 

C_0 + C_1 + C_2 + .........+C_n = 2^n

-

Now,

\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}+\cdots+\mathrm{C}_{n}\right)^3=2^{3n}

And

\;1+^{3 n} \mathrm{C}_{1}+^{3 n} \mathrm{C}_{2}+\cdots+^{3 n} \mathrm{C}_{3 n}

=\;^{3n}\mathrm{C}_0+^{3 n} \mathrm{C}_{1}+^{3 n} \mathrm{C}_{2}+\cdots+^{3 n} \mathrm{C}_{3 n} = 2^{3n}

Hence, Option B is correct.

Posted by

sudhir.kumar

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