Write down $\overrightarrow{\text{PA}}$ and $\overrightarrow{\text{PB}}$.

Write down the value of $\lambda$ corresponding to the point $\text{A}$.

the girls do not sit on either end and do not sit together.

the girls do not sit together.

Find how many sets of three points can be selected which can form the vertices of a triangle.

The third term in the expansion is the mean of the second term and the fourth term in the expansion.

Find the possible values of $x$.

Determine the mean of X.

Find the term independent of $x$ in the expansion of $\frac{1}{x^3}{\left(\frac{1}{3x^2}-\frac{x}{2}\right)}^9$.

Consider the expansion of ${\left(2+x\right)}^n$, where $n⩾3$ and $n\in \mathbb{Z}$.

The coefficient of $x^3$ is four times the coefficient of $x^2$. Find the value of $n$.

In a trial examination session a candidate at a school has to take 18 examination papers including the physics paper, the chemistry paper and the biology paper. No two of these three papers may be taken consecutively. There is no restriction on the order in which the other examination papers may be taken.

Find the number of different orders in which these 18 examination papers may be taken.

Express the binomial coefficient $\left(\begin{array}{c}3n+1\\ 3n-2\end{array}\right)$ as a polynomial in $n$.

Let $f(x)=(x^2+3{)}^7$. Find the term in $x^5$ in the expansion of the derivative, $f^{\prime }(x)$.

Hence find the least value of $n$ for which $\left(\begin{array}{c}3n+1\\ 3n-2\end{array}\right)>{10}^6$.

In the expansion of $(x+k{)}^7$, where $k\in \mathrm{ℝ}$, the coefficient of the term in $x^5$ is $63$.

Find the possible values of $k$.

Let $\text{C}$ be the point on $l_1$ with coordinates (1, 0, 1) and $\text{D}$ be the point on $l_2$ with parameter $\mu =-2$.

Find the area of the quadrilateral $\text{CDBA}$.

Show that the probability that Chloe wins the game is $\frac{3}{8}$.

Find the value of $a$.

There are more males than females in the group.

the girls do not sit on either end.

Find the number of different possible teams that could be chosen, given that the team must include at least one girl and at least one boy.

Consider the expansion of ${\left(3x^2-\frac{k}{x}\right)}^9$, where $k>0$.

The coefficient of the term in $x^6$ is $6048$. Find the value of $k$.

Two of the teachers, Gary and Gerwyn, refuse to go out for a meal together.

Consider the expansion of ${\left(8x^3-\frac{1}{2x}\right)}^n$ where $n\in {\mathrm{\mathbb{Z}}}^{+}$. Determine all possible values of $n$ for which the expansion has a non-zero constant term.

Expand and simplify ${(1-a)}^3$ in ascending powers of $a$.

Show that $b=21$.

Verify that $\text{P}$ is the point of intersection of the two lines.

Determine the variance of X.

Three girls and four boys are seated randomly on a straight bench. Find the probability that the girls sit together and the boys sit together.

Hence or otherwise find the coefficient of the term in $x^9$ in the expansion of ${\left(x+3\right)}^{11}$.

Use the binomial theorem to expand ${\left(\cos \theta +\mathrm{i} \sin \theta \right)}^4$. Give your answer in the form $a+b\mathrm{i}$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

The coefficient of $x^2$ in the expansion of ${\left(\frac{1}{x}+5x\right)}^8$ is equal to the coefficient of $x^4$ in the expansion of ${\left(a+5x\right)}^7,a\in \mathbb{R}$. Find the value of $a$.

Consider the expansion of $(3+x^2{)}^{n+1}$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Given that the coefficient of $x^4$ is $20 412$, find the value of $n$.