Binomial theorem practice · Easy

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 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 ($2^{{18}}$) 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.

Option A (0) is correct because the sum of the series is zero due to the alternating signs and the properties of binomial coefficients.

Option B (210) is correct because the coefficient of ${x^{ - 10}}$ can be found by determining the correct term in the binomial expansion.

Option B ($- \frac{{896}}{{27}}$) is correct because it is the 6th term in the expansion of the given binomial expression, calculated using the binomial theorem.

Option C ($\frac{1}{4} n(n+1)(n+2)(n+3)$) is correct because it represents the sum of the series 1.2.3 + 2.3.4 + 3.4.5 + ... up to n terms.

Option C (23) is correct because when n=23, the coefficients of the 9th, 10th, and 11th terms of (1+x)^n are in arithmetic progression.

Option B (476) is correct because the coefficient of $x^{10}$ in the expansion of $(1+x^2-x^3)^8$ can be found by considering the relevant terms in the multinomial expansion.

The correct answer is D (729) because the sum of the coefficients is found by substituting x = 1 into the expansion.

Option C (${\;^{24}}{C_{12}}$) is correct because in the expansion of ${\left( {1 + x} \right)^{24}}$, the largest coefficient is the middle term, which is the 13th term, and its coefficient is ${\;^{24}}{C_{12}}$.

Option B (p=2, n=12) is correct. The values of p and n are determined by comparing the coefficients of x and x² in the binomial expansion.

Option B ($(n+1)$) is correct because the coefficient of $x^n$ in the expansion of $(1-x)^{-2}$ is indeed $n+1$.

Option D ($A^2-B^2$) is correct because it directly results from multiplying (A+B) and (A-B), which represent (x+a)^n and (x-a)^n respectively.

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the rth term from the end is indeed the (n-r+2)th term, and the assertion correctly states the value of that term.

Option B (2) is correct because the remainder when 8⁽²ⁿ⁾ - (62)⁽²ⁿ⁺¹⁾ is divided by 9 is 2.

Option B ( $(-1)^{n} \, \, {}^{2 n} C_n$ ) is correct because the coefficient of the term independent of $x$ is found by simplifying the expression and finding the coefficient of $x^{2n}$ in the resulting expansion.

Option A ($\frac{{{3^n} + 1}}{2}$) is correct because it is derived by substituting x = 1 and x = -1 into the given equation and solving for the required sum.

Option A ($ab = 1$) is correct because equating the coefficients of ${x^7}$ and ${x^{ - 7}}$ leads to this relationship between $a$ and $b$.

Option B (3) is correct because the remainder is found by using the binomial theorem and focusing on the remainder when divided by 80.