Binomial theorem practice · Mixed difficulty

Option B (${ }^{10} C_4$) is correct because it represents the coefficient of $x^{1012}$ obtained by finding the appropriate values of r and s in the multinomial expansion.

Option C ($\frac{{405}}{{256}}$) is correct because after finding the general term and substituting the correct value of r (r=2), the coefficient of x⁴ simplifies to this fraction.

Option A ($\frac{1}{2}(n+1)(n+2)$) is correct because the number of terms in the expansion of $(x+y+z)^n$ is equivalent to the number of non-negative integer solutions to the equation $p+q+r=n$, which is given by ${n+3-1 \choose 3-1} = {n+2 \choose 2} = \frac{(n+2)(n+1)}{2}$.

The correct answer is C (12.25) because it correctly calculates the ratio of a₂ to a₀ using the binomial theorem expansion of the given expression.

Option A (0) is correct because the given summation simplifies to 0 using binomial theorem and algebraic manipulation.

Option B ($-\frac{1}{n}$) is correct because when n is odd, the coefficient of ${x^n}$ in the expansion of ${\log _e}(\frac{1}{{1 + x + {x^2} + {x^3}}})$ is given by -1/n.

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the number of terms in the given expansion is indeed 6, and the reason provides the correct formula for the number of terms when n is even.

Option C ($2(-1)^{n-1} n^{2 n-1} C_n$) is correct because it is derived by differentiating the binomial expansion and comparing coefficients.

Option B, 5456, is correct because it represents the number of non-negative integral solutions to the inequality.

Option A ($\frac{1}{2}(n+1)(n+2)$) is correct because it represents the number of non-negative integral solutions of the equation p+q+r=n, which corresponds to the number of terms in the multinomial expansion.

Option B ($\frac{{2 n !}}{{(n-1) !(n+1) !}}$) is correct because it represents the coefficient of x^(n-1) in the expansion of (1+x)⁽²ⁿ⁾, which corresponds to the coefficient of 1/x in the original expression.

Option D (198) is correct because it is the coefficient of the term independent of x after expanding and simplifying the given expression.

Option B (14) is correct because equating the binomial coefficients leads to two possible values for r, and 14 is one of them.

Option D (43) is correct because the sum of the rational terms in the binomial expansion occurs when r=0 and r=12, resulting in 27+16=43.

Option A (5/3) is correct because it is the coefficient of the term independent of x after expanding the given binomial expression and solving for the term where the power of x is zero.

Option D (63) is correct because by expanding the expression and equating the coefficients of x and x², we can solve for p and q, and then calculate 2p + 3q.

Option A ($n[2^{2 n-1}-{ }^{2 n-1} C_{n-1}]$) is correct because it correctly uses the symmetry and properties of binomial coefficients to simplify the summation.

Option B (1024) is correct because it is the coefficient of $x^7$ obtained by applying the binomial theorem to $(1-2x)^{-2}$.

Option C (33) is correct because the number of integral terms is found by determining the values of r for which both exponents are integers.

Option D (3) is correct because after simplification, the highest power of x in the polynomial is 3.

Option A, $\frac{(n+1)(n+2)}{2}$, is correct because it represents the number of ways the student can score 2n marks in three subjects, each with a maximum of n marks.

Option C (9) is correct because after simplification using the binomial theorem, the highest power of x is 9.

Option A (${ }^{300} C_{50}$) is correct because the summation represents the coefficient of $x^{50}$ in the product of two binomial expansions, which can be simplified using the Vandermonde identity.

Option D (None of these) is correct because $2^{3n}-1$ is divisible by 7, not 8, 16, or 32.

Option A ($2^{{18}}$) is correct because the sum of the last ten coefficients in the expansion of $(1+x)^{{19}}$ is $2^{{18}}$, which is found using binomial properties.

Option A (0) is correct because when $3^{100}$ is expressed as $(10-1)^{50}$, the hundreds digit is determined to be 0 after binomial expansion.

Option A (4) is correct because the value of $a^2$ is 4, which is calculated by finding the term independent of x in the binomial expansion.

Option C (-13720) is correct because the term independent of x is found using the multinomial theorem, and the correct combination of powers results in -13720.

Option C (128) is correct because the sum of binomial coefficients in the expansion of $(1+x)^7$ is obtained by setting $x=1$, resulting in $(1+1)^7 = 2^7 = 128$.

Option A (3) is correct because the remainder is found by calculating the sum of factorials up to 4! since all factorials greater than 4! are divisible by 15.