Recent questions tagged binomial-theorem

Use question $14$ and Corollary $1$ to show that if $n$ is an integer greater than $1,$ then $\binom{n}{\left \lfloor n/2 \right \rfloor}\geq \frac{2^{n}}{2}.$Conclude fr...

Show that if $n$ is a positive integer, then $1 = \binom{n}{0}<\binom{n}{1}<\dots < \binom{n}{\left \lfloor n/2 \right \rfloor} = \binom{n}{\left \lceil n/2 \right \rceil...

What is the row of Pascal’s triangle containing the binomial coefficients $\binom{9}{k} ,\: 0 \leq k \leq 9?$

The row of Pascal’s triangle containing the binomial coefficients $\binom{10}{k},\: 0 \leq k \leq 10, \:\text{is:}\: 1\:\: 10\:\: 45\:\: 120\:\: 210\:\: 252\:\: 210\:\: 1...

Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x^{2} − \frac{1}{x}\right)^{100},$ where $k$ is an integer.

Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x + \frac{1}{x}\right)^{100},$ where $k$ is an integer.

What is the coefficient of $x^{101}y^{99}$ in the expansion of $(2x − 3y)^{200}?$

What is the coefficient of $x^{8}y^{9}$ in the expansion of $(3x + 2y)^{17}?$

What is the coefficient of $x^{9}\:\text{in}\: (2 − x)^{19}?$

What is the coefficient of $x^{7}\:\text{in}\: (1 + x)^{11}?$

How many terms are there in the expansion of $(x + y)^{100}$ after like terms are collected?

Find the coefficient of $x^{5}y^{8}\:\text{in}\: (x + y)^{13}.$

Find the expansion of $(x + y)^{6}.$

Find the expansion of $(x + y)^{5}$using combinatorial reasoning, as in Example $1.$using the binomial theorem.

Find the expansion of $(x + y)^{4}$using combinatorial reasoning, as in Example $1.$ using the binomial theorem.

Let $(1+x)^n = C_0+C_1x+C_2x^2+ \dots + C_nx^n$, $n$ being a positive integer. The value of$$\left( 1+\dfrac{C_0}{C_1} \right) \left( 1+\dfrac{C_1}{C_2} \right) \cdots \l...

$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals$2^n+n2^{n-1}$$2^n-n2^{n-1}$$2^n$none of these

The following sum of $n+1$ terms $$2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \en...

Let $(1+x)^n = C_0+C_1x+C_2x^2+ \ldots +C_nx^n, \: n$ being a positive integer. The value of $$\left( 1+\frac{C_0}{C_1} \right) \left( 1+\frac{C_1}{C_2} \right) \cdots \...

The value of $(1.1)^{10}$ correct to $4$ decimal places is$2.4512$$1.9547$$2.5937$$1.4512$

The value of the term independent of $x$ in the expansion of $(1-x)^2(x+\frac{1}{x})^7$ is$-70$$70$$35$None of these

The value of the term independent of $x$ in the expansion of $(1-x)^{2}(x+\frac{1}{x})^{7}$ is$-70$$70$$35$None of these

The coefficient of $x^6y^3$ in the expression $(x+2y)^9$ is$84$$672$$8$none of these

The value of $^{13}C_{3} + ^{13}C_{5} + ^{13}C_{7} +\dots + ^{13}C_{13}$ is$4096$$4083$$2^{13}-1$$2^{12}-1$

The number of terms independent of $x$ in the binomial expansion of $\left(3x^2 + \dfrac{1}{x}\right)^{10}$ is $0$$1$$2$$5$

Which power of x has the greatest coefficient in the expansion of (1+1/2 x)^10 ?