\section*{6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
The cancellation of the Wimbledon tournament has led to a world surplus of tennis balls, and Santa has decided to use them as stocking fillers. He comes down the chimney with $n$ identical tennis balls, and he finds $k$ named stockings waiting for him.

Let $g ( n , k )$ be the number of ways that Santa can put the $n$ balls into the $k$ stockings; for example, $g ( 2,2 ) = 3$, because with two balls and two children, Miriam and Adam, he can give both balls to Miriam, or both to Adam, or he can give them one ball each.\\
(i) What is the value of $g ( 1 , k )$ for $k \geqslant 1$ ?\\
(ii) What is the value of $g ( n , 1 )$ ?\\
(iii) If there are $n \geqslant 2$ balls and $k \geqslant 2$ children, then Santa can either give the first ball to the first child, then distribute the remaining balls among all $k$ children, or he can give the first child none, and distribute all the balls among the remaining children. Use this observation to formulate an equation relating the value of $g ( n , k )$ to other values taken by $g$.\\
(iv) What is the value of $g ( 7,5 )$ ?\\
(v) After the first house, Rudolf reminds Santa that he ought to give at least one ball to each child. Let $h ( n , k )$ be the number of ways of distributing the balls according to this restriction. What is the value of $h ( 7,5 )$ ?



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