Moderate -0.8 This is a straightforward application of the binomial theorem requiring only standard recall and basic algebraic manipulation. Part (a) is direct expansion using binomial coefficients, part (b) involves equating two terms and solving a simple equation, and part (c) applies the binomial probability formula to verify equality—all routine AS-level techniques with no problem-solving insight required.
Write down and simplify the first four terms in the expansion of \(( x + y ) ^ { 7 }\).
Give your answer in ascending powers of \(x\).
Given that the terms in \(x ^ { 2 } y ^ { 5 }\) and \(x ^ { 3 } y ^ { 4 }\) in this expansion are equal, find the value of \(\frac { x } { y }\).
A hospital consultant has seven appointments every day.
The number of these appointments which start late on a randomly chosen day is denoted by \(L\).
The variable \(L\) is modelled by the distribution \(\mathrm { B } \left( 7 , \frac { 3 } { 8 } \right)\).
Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.
10
\begin{enumerate}[label=(\alph*)]
\item Write down and simplify the first four terms in the expansion of $( x + y ) ^ { 7 }$.\\
Give your answer in ascending powers of $x$.
\item Given that the terms in $x ^ { 2 } y ^ { 5 }$ and $x ^ { 3 } y ^ { 4 }$ in this expansion are equal, find the value of $\frac { x } { y }$.
\item A hospital consultant has seven appointments every day.
The number of these appointments which start late on a randomly chosen day is denoted by $L$.\\
The variable $L$ is modelled by the distribution $\mathrm { B } \left( 7 , \frac { 3 } { 8 } \right)$.
Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.
\end{enumerate}
\hfill \mbox{\textit{OCR AS Pure 2017 Q10 [7]}}