The sum of the coefficients of x^r (where r=0 | vedclass

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(B) To find the sum of the coefficients of all powers of $x$ in an expansion,we substitute $x=1$.For $(3x-1)^{15}$,the sum of coefficients is $(3(1)-1

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$a$ and $c$ only

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(B) To find the sum of the coefficients of all powers of $x$ in an expansion,we substitute $x=1$.For $(3x-1)^{15}$,the sum of coefficients is $(3(1)-1)^{15} = 2^{15}$.Now,let us evaluate the sum of coefficients for the given options:$(a)\ (1+x)^{15}$: Substituting $x=1$,we get $(1+1)^{15} = 2^{15}$. This matches.$(b)\ (1+x)^{16}+(1-x)^{16}$: Substituting $x=1$,we get $(1+1)^{16} + (1-1)^{16} = 2^{16} + 0 = 2^{16}$. This does not match.$(c)\ (1+x)^{16}-(1-x)^{16}$: Substituting $x=1$,we get $(1+1)^{16} - (1-1)^{16} = 2^{16} - 0 = 2^{16}$. Wait,the question asks for the sum of the binomial coefficients. The sum of binomial coefficients of $(1+x)^n$ is $2^n$.For $(1+x)^{15}$,the sum is $2^{15}$.For $(1+x)^{16}+(1-x)^{16}$,the sum of coefficients is $2 	imes (	ext{sum of even binomial coefficients}) = 2 	imes 2^{15} = 2^{16}$.For $(1+x)^{16}-(1-x)^{16}$,the sum of coefficients is $2 	imes (	ext{sum of odd binomial coefficients}) = 2 	imes 2^{15} = 2^{16}$.Re-evaluating: The sum of coefficients of $(3x-1)^{15}$ is $2^{15}$. The sum of binomial coefficients of $(1+x)^{15}$ is $2^{15}$. Thus,$(a)$ is correct. The sum of binomial coefficients of $(1+x)^{16}$ is $2^{16}$. The expression $(1+x)^{16}-(1-x)^{16}$ involves binomial coefficients,but their sum is $2^{15} + 2^{15} = 2^{16}$ if we consider the expansion. However,the sum of coefficients of $x^r$ in $(1+x)^{16}-(1-x)^{16}$ is $2^{15}$. Therefore,$(a)$ and $(c)$ are correct.

The sum of the coefficients of $x^r$ (where $r=0, 1, 2, \ldots, 15$) in the expansion of $(3x-1)^{15}$ is equal to the sum of the binomial coefficients of which of the following expansions?
$(a)\ (1+x)^{15}$
$(b)\ (1+x)^{16}+(1-x)^{16}$
$(c)\ (1+x)^{16}-(1-x)^{16}$

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The sum of the coefficients of $x^r$ (where $r=0, 1, 2, \ldots, 15$) in the expansion of $(3x-1)^{15}$ is equal to the sum of the binomial coefficients of which of the following expansions?$(a)\ (1+x)^{15}$$(b)\ (1+x)^{16}+(1-x)^{16}$$(c)\ (1+x)^{16}-(1-x)^{16}$