5. For ALL APPLICANTS.

A triangular triple is a triple of positive integers ( $a , b , c$ ) such that we can construct a triangle with sides of length $a , b$ and $c$. This means that the sum of any two of the numbers is strictly greater than the third; so if $a \leqslant b \leqslant c$, then it is equivalent to requiring $a + b > c$. For example, ( $3,3,3$ ) and ( $4,5,3$ ) are triangular triples, but ( $1,3,2$ ) and ( $3,3,6$ ) are not. For any positive integer $P$, we define $f ( P )$ to be the number of triangular triples such that the perimeter $a + b + c$ is equal to $P$. Triples with the same numbers, but in a different order, are counted as being distinct. So $f ( 12 ) = 10$, because there are 10 triangular triples with perimeter 12, shown below:

$( 3,4,5 )$	$( 3,5,4 )$	$( 4,3,5 )$	$( 4,5,3 )$	$( 5,3,4 )$	$( 5,4,3 )$
$( 2,5,5 )$	$( 5,2,5 )$	$( 5,5,2 )$
$( 4,4,4 )$

(i) Write down the values of $f ( 3 ) , f ( 4 ) , f ( 5 )$ and $f ( 6 )$.
(ii) If ( $a , b , c$ ) is a triangular triple, show that ( $a + 1 , b + 1 , c + 1$ ) is also a triangular triple.
(iii) If ( $x , y , z$ ) is a triangular triple, with $x + y + z$ equal to an even number greater than or equal to 6 , show that each of $x , y , z$ is at least 2 and that $( x - 1 , y - 1 , z - 1 )$ is also a triangular triple.
(iv) Using the previous two parts, prove that for any positive integer $k \geqslant 3$,
$$f ( 2 k - 3 ) = f ( 2 k )$$
(v) We will now consider the case where $P \geqslant 6$ is even, and we will write $P = 2 S$.
(a) Show that in this case ( $a , b , c$ ) is a triangular triple with $a + b + c = P$ if and only if each of $a , b , c$ is strictly smaller than $S$.
(b) For any $a$ such that $2 \leqslant a \leqslant S - 1$, show that the number of possible values of $b$ such that $( a , b , P - a - b )$ is a triangular triple is $a - 1$. Hence find an expression for $f ( P )$ for any even $P \geqslant 6$.
(vi) Find $f ( 21 )$.
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Answer the following questions.
(1) Calculate the number of possible ways to distribute $n$ equivalent balls to $r$ distinguishable boxes such that each box contains at least one ball, where $n \geq 1$ and $1 \leq r \leq n$.
Next, consider to place $n$ black balls and $m$ white balls in a line uniformly at random. A run is defined to be a succession of the same color. Let $r$ be the number of runs of black balls and $s$ be the number of runs of white balls. Assume that $n \geq 1 , m \geq 1,1 \leq r \leq n$, and $1 \leq s \leq m$.
(2) Calculate the total number of arrangements when we do not distinguish among balls of the same color.
(3) Calculate the probability $P ( r , s )$ that the number of runs of black balls is $r$ and the number of runs of white balls is $s$.
(4) Calculate the probability $P ( r )$ that the number of runs of black balls is $r$.
(5) Using $( 1 + x ) ^ { n } ( 1 + x ) ^ { m } = ( 1 + x ) ^ { n + m }$, show that the following equations hold.
$$\begin{aligned} \sum _ { \ell = 0 } ^ { \min \{ n , m \} } \binom { n } { \ell } \binom { m } { \ell } & = \binom { n + m } { m } \\ \sum _ { \ell = 0 } ^ { \min \{ n - 1 , m \} } \binom { n } { \ell + 1 } \binom { m } { \ell } & = \binom { n + m } { m + 1 } \end{aligned}$$
(6) Calculate the expected value $E ( r )$ and the variance $V ( r )$ of $r$.
Calculate $\lim _ { N \rightarrow \infty } \frac { E ( r ) } { N }$ and $\lim _ { N \rightarrow \infty } \frac { V ( r ) } { N }$ supposing that $N = n + m$ and $\lim _ { N \rightarrow \infty } \frac { n } { N } = \lambda$, where $\lambda$ is a real constant.