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.