Binomial theorem practice · Mixed difficulty

Option C (64) is correct because by expressing ${7^9}$ and ${9^7}$ as $(8-1)^9$ and $(8+1)^7$ respectively and using the binomial theorem, we can show that their sum is divisible by 64.

Option D (${ }^{45} \mathrm{C}_{23}$) is correct because the sum can be identified as the coefficient of $x^{23}$ in the expansion of $(1+x)^{22}(1+x)^{23}$.

Option B (${}^{4 n} C_{2 n}$) is correct because the product of binomial coefficients cannot exceed the largest binomial coefficient in the expansion of $(1+x)^{4n}$.

Option C ($n(n-1) \cdot 2^{n-3}$) is correct because it is derived by simplifying the double summation involving binomial coefficients.

Option A (5) is correct because after expanding and simplifying the expression, only the even power terms remain, resulting in 5 terms.

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 (210) is correct because after finding n=10, the 5th term can be calculated using the binomial theorem.

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.

The correct answer is C (${n^2}$) because the infinite sum can be simplified to the derivative of a geometric series and evaluated.

Option A (405/256) is correct because by finding the general term in the binomial expansion and setting the power of x to 4, we can determine the coefficient of x⁴.

Option B ($\frac{1}{11}(3^{11}-1)$) is correct because the given expression can be represented as an integral of a binomial expansion.

Option C (5) is correct because when expanding the given expression, only the even powers of $x$ remain, resulting in 5 non-zero terms.

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 B ($\int_0^1\left(1-x^2\right)^n d x$) is correct because the given series can be represented as an integral of $(1-x^2)^n$ from 0 to 1.

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 D ($\sqrt{\frac{5}{3}}$) is correct because the sum of the series can be found by recognizing it as a binomial expansion.

Option D (3) is correct because substituting x=1 into the polynomial gives the sum of the coefficients, which is then solved for a.

Option A ($^{501}{C_{101}}{(5)^{399}}$) is correct because by simplifying the expression and using binomial theorem, the coefficient of x¹⁰¹ can be found.

Option C (${}^{102} C_{50}$) is correct because the given series simplifies to $(1+x)^{102} - x^{101}(1+x) - 101x^{101}$, and the coefficient of $x^{50}$ is ${}^{102} C_{50}$.

Option C (0) is correct because the given expression is the binomial expansion of (1 + x)^n with x = -1, which results in 0.

Option D (540) is correct because by finding the general term in the expansion and setting the power of x to 3, we can find the coefficient of $x^3$ to be 540.

Option C (30,31) is correct because by equating the coefficients of two consecutive terms in the binomial expansion of ${\left( {3 + 2x} \right)^{74}}$, we find that the terms are the 30th and 31st terms.

Option A (18) is correct because it is the coefficient of ${x^5}$ obtained by expanding the binomial expression ${\left( {x + 3} \right)^6}$ using the binomial theorem.

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, based on the relationship between arithmetic and geometric means, the expression is always greater than zero.