Binomial Theorem
Sum of the coefficients of $$ (1+x)^n $$ is always a
Find the sum of coefficients in the expansion of $$\left(1-\dfrac 2x+\dfrac {4}{x^2}\right)^n$$ given that the number of terms are $$28$$
$$\left(1-\dfrac 2x+\dfrac {4}{x^2}\right)^n$$ and number of terms $$=28$$
Number of terms $$=^{n+3-1}C_{3-1} = ^{n+2}C_2$$
$$\dfrac {(n+2)!}{2! (n!)} = \dfrac {(n+2)(n+1)}{2} = 28$$
$$n^2+3n-54=0$$
$$n^2-6n+9n-54=0$$
$$(n+3)(n-6)=0$$
$$n=6$$
Number of terms $$=\left(1-\dfrac 2x+\dfrac {4}{x^2}\right)^6$$
Sum of coefficients is known when $$x=1$$
No. of terms $$=(1-2+4)^6 = 3^6 = 729$$
Sum of the coefficients of $$ (1+x)^n $$ is always a
The sum of the coefficients of even powers of $$x$$ in the expansion of $$ (1+x+x^2+x^3)^5 $$ is
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The value of $$\displaystyle \sum_{r = 1}^{15} r^{2} \left (\dfrac {^{15}C_{r}}{^{15}C_{r - 1}}\right )$$ is equal to:
The sum of coefficient of integral powers of $$x$$ in the binomial expansion of $$(1-2\sqrt x)^{50}$$ is :
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of $$(x + a)^{n}$$ are $$A$$ and $$B$$ respectively, then the value of $$(x^{2} - a^{2})^{n}$$ is
Let $$X = (^{10}C_{1})^{2} + (^{10}C_{2})^{2} + 3(^{10}C_{3})^{2} + .... + 10 (^{10}C_{10})^{2}$$, where $$^{10}C_{r}, r\epsilon \left \{1, 2, ..., 10\right \}$$ denote binomial coefficients. Then, the value of $$\dfrac {1}{1430}X$$ is
The sum of the coefficients of even powers of $$x$$ in the expansion of $$ (1+x+x^2+x^3)^5 $$ is