Binomial theorem practice · Medium

The correct answer is A (${\;^{2n}}{C_n}$) because it is found by simplifying the expression and identifying the middle term in the binomial expansion.

Option C (9) is correct because the remainder is found by expressing 27⁴⁰ as a power of 3 and using the binomial theorem.

Option B ($a_{2n-r}, 0 \leq r \leq 2n$) is correct because the coefficients of the polynomial are symmetric.

Option A ($c^n$) is correct because the summation can be expressed as the real part of a complex number raised to the power of n, which simplifies to $c^n$ using the Law of Cosines.

Option A (0) is correct because both S and T are equal to the Beta function B(m+1, n+1), making S-T = 0.

Option A is correct. By substituting x=1 and x=-1 into the given equation and adding the resulting equations, we isolate the desired sum.

Option A (^{16}C_9) is correct because the coefficient of x⁶ in the given sum of binomial expansions is found by using the formula for the sum of a geometric series and then finding the coefficient of x⁷ in the resulting expression.

Option A (0) is correct because the exponent of x in the general term of the binomial expansion cannot be equal to 6 for any integer value of r.

Option B ($\sum\limits_{{r=0}}^n C_r^3$) is correct because the coefficient of $x^n y^n$ is obtained by multiplying the coefficients of $x^r$, $y^{n-r}$ and $x^{n-r}y^r$ and summing over all possible values of $r$.

Option C ($150 a$) is correct because by using the property $a_r = a_{300-r}$, we can simplify the summation and find the required value.

Option B (0) is correct because the given expression simplifies to 0 using binomial coefficient properties.

Option A (15) is correct because it is the coefficient of $t^4$ in the expansion of the given expression, obtained by using the binomial theorem.

Option A ($\frac{3^n+1}{2}$) is correct because by substituting x=1 and x=-1 into the given equation, we can eliminate the odd-powered terms and solve for the sum of the even-powered terms.

Option B (18) is correct because for the term to be independent of x, 2n must be a multiple of 3, making n a multiple of 3/2. Among the options, only 18 satisfies this condition.

Option D (210) is correct because it represents the greatest coefficient in the multinomial expansion, which occurs when the powers are as close as possible.

Option C (${ }^{x+y} C_r$) is correct because the given expression is the coefficient of $a^r$ in the expansion of $(1+a)^x (1+a)^y = (1+a)^{x+y}$.

Option D (3) is correct because (1 + 0.0001)¹⁰⁰⁰⁰ approaches e, which is approximately 2.718, and the next integer greater than that is 3.

Option C is correct because the expression simplifies to the given result using combinatorial identities and properties.

The correct answer is 2,6 because substituting these values into the binomial expansion allows us to match the coefficient of the 4th term to the given value of 20.

Option B ($\frac{{n\left( {n - 1} \right) \ldots \left( {n - r + 2} \right)}}{{\left( {r - 1} \right)!}}{a^{n - r + 1}}{\left( {2x} \right)^{r - 1}}$) is correct because it correctly represents the rth term in the binomial expansion of the given expression.

Option B (8) is correct because when ${5^{99}}$ is divided by 13, the remainder is 8, which is obtained by breaking down ${5^{99}}$ and using properties of remainders.

Option B ($\frac{\mathrm{\pi }}{3}$) is correct because it satisfies the given ratio of the 4th term to the 2nd term in the expansion.

Option A ($\frac{{15 !}}{{3 !(4 !)^3}}$) is correct because it represents the greatest multinomial coefficient when the power is divided as evenly as possible among the variables.

Option B ($\frac{{(2 n+1)^{2 n}}}{{(2 n) !}}$) is correct because it is obtained by simplifying the product of terms involving binomial coefficients.

Option C, 102, is correct because it represents the number of terms where both exponents in the binomial expansion are integers.

Option B (${2^n}$) is correct because the coefficient of ${x^n}$ in the expansion of $\frac{{{{\left( {1 + x} \right)}^n}}}{{1 - x}}$ is the sum of all coefficients in the expansion of ${(1+x)^n}$, which is ${2^n}$.

Option C ($3k$) is correct because the given condition holds true only when $n$ is a multiple of 3, which can be represented as $3k$.

Option D (0) is correct because the given expression simplifies to zero due to the properties of binomial coefficients and alternating signs.

Option C ($\frac{1}{\sqrt{2}}$) is correct because it is the eccentricity of the ellipse formed by the locus of the midpoint, derived from the original ellipse's equation.

Option D ($[P]+[Q]=$ Even) is correct because by analyzing the binomial expansion and conjugate terms, we can determine that both [P] and [Q] are even.