TS EAMCET · Maths · Binomial Theorem

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

A $(a), (b)$ and $(c)$
B $(a)$ and $(c)$ only
C $(b)$ and $(c)$ only
D $(a)$ and $(b)$ only

Verified Solution

Answer & Solution

Correct Answer

(B) $(a)$ and $(c)$ only

Step-by-step Solution

Detailed explanation

The binomial expansion of a+bn=C0nanb0+C1nan-1b1+C2nan-2b2+...+Cnna0bn and to find the sum of coefficients of all the powers of x, put x=1, in the expansion. Hence, the sum of coefficients of all the powers of x, in 3x-115 is 3×1-115=215. Also, the sum of coefficients of…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Sum of the coefficients of xr, (r=0, 1, 2, 3,..., 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

Step-by-step Solution

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