Binomial theorem practice · Easy

Option D ($3^n-2^n$) is correct because the double summation can be simplified using combinatorial identities to this expression.

Option B (3.2) is correct because it is the regression coefficient of x on y, calculated using the given correlation coefficient and regression coefficient of y on x.

The correct answer is B (${2^{58}}$) because the sum of the last 30 coefficients is half of the total sum of all coefficients in the binomial expansion.

Option B (${x^2}$) is correct because the expansion of ${\left( {1 + x} \right)^n} - nx - 1$ contains terms with powers of $x$ starting from ${x^2}$, making it divisible by ${x^2}$.

Option D (64) is correct because ${49^n} + 16n - 1$ can be rewritten as ${(1 + 48)^n} + 16n - 1$, and using the binomial theorem, it can be shown that the expression is divisible by 64.

Option D (8) is correct because when 2⁴⁰³ is divided by 15, the remainder is 8, making the fractional part 8/15.

Option B is correct. The coefficient of x^n in the expansion is found by summing the coefficients of x^n and x^(n-1) in (1-x)^n.

Option A ($\frac{1}{{2\sqrt 3 }}$) is correct because it is the positive value of 'a' that makes the coefficients of x⁵ and x¹⁵ equal in the given binomial expansion.

Option A (nx) is correct because the series can be recognized as the derivative of a binomial expansion.

Option D (None of these) is correct because the given sum can be simplified to ${2^{39}} + \frac{1}{2} \times \frac{{40!}}{{{{\left( {20!} \right)}^2}}}$, which does not match any of the provided options.

Option B (a negative integer) is correct because the expression simplifies to -1, which is a negative integer.

Option D (-455) is correct because it is the coefficient of x³⁹ in the binomial expansion of the given expression, calculated using the general term formula.

Option D (an irrational number) is correct because the expression simplifies to $2\sqrt{3}$ multiplied by an integer, making it an irrational number.

Option A ($a_p=a_q$) is correct because the binomial coefficients ${}^{p+q}C_p$ and ${}^{p+q}C_q$ are equal due to the property ${}^nC_r = {}^nC_{n-r}$.

Option D ($400\,{\,^{38}}{C_{20}}$) is correct because the given summation can be simplified using combinatorial identities to a form involving ${ }^{38} C_{20}$.

Option A ($(n+1) 2^{n+1}$) is the correct answer because the summation can be simplified by separating the terms and using the binomial theorem.

Option B ($r+1$) is correct because the given expression is the expansion of $(1-x)^{-2}$, and the coefficient of $x^r$ is $r+1$.

Option D (${\;^n}{C_4} + {\;^n}{C_2} + {\;^n}{C_1} \times {\;^n}{C_2}$) is correct because it represents the sum of the products of binomial coefficients that result in the $x^4$ term when expanding the given expression.

Option B ($(2 {a}+{nb}) 2^{{n}-1}$) is correct because the given series can be simplified using binomial coefficient properties and summation techniques.

Option B (924) is correct because when the sum of coefficients in ${\left( {x + y} \right)^n}$ is 4096, n is 12, and the greatest coefficient is ${\;^{12}}{C_6}$ which equals 924.

Option A (${\;^{60}}{C_{40}}\;\;$) is correct because ${(1 + 3x + 3{x^2} + {x^3})^{20}}$ simplifies to ${(1 + x)^{60}}$, and the coefficient of ${x^{20}}$ is ${\;^{60}}{C_{20}}$ which is equal to ${\;^{60}}{C_{40}}$.

Option A ($2^{2n}$) is correct because the summation can be simplified to the square of the sum of binomial coefficients, which equals $2^{2n}$.

Option C (141) is correct because the constant term in the expansion is obtained by summing the products of coefficients that result in $x^0$.

Option A (4) is correct because after simplification, the coefficient of $x^{13}$ in the expansion is found to be 4.

Option D (430) is correct because it represents the number of ways to distribute the candies, calculated using the multinomial theorem and considering the given constraints on the number of candies each child can receive.

Option D (2) is correct because the greatest value of the term independent of x is achieved when sin(2α)=1, leading to a=2 after solving the equation.

Option D (8) is correct because when $(11)^{1011}+(1011)^{11}$ is divided by 9, the remainder is equivalent to the remainder of $2^{1011}+3^{11}$ divided by 9, which simplifies to 8.

Option A (0) is correct because after finding the values of p, which are $\pm 1$ and $\pm \sqrt{3}$, their sum is 0.

Option B ($n(n+1)$) is correct because the given expression simplifies to $2 \sum_{r=1}^n(n-r+1)$, which equals $n(n+1)$.

Option B (-n) is correct because it is derived by differentiating the given expression and substituting x = -1.