Question 89
Set $f(n, k)$ as the number of ways to distribute $k$ chocolates for $n$ children where each child receives 0, 1 or 2 chocolates. For example, $f(3,4)=6$, $f(3,6)=1$ and $f(3,7)=0$.
a) Display all 6 ways to distribute 4 chocolates to 3 children with each one earning a maximum of two chocolates.
b) Considering 2015 children, check that $f(2015, k)=0$ for all $k$ greater than or equal to an appropriate value.
c) Shows that the equation $$
f(2016, k)=f(2015, k)+f(2015, k-1)+f(2015, k-2)
$$ is true for all $k$ positive integer greater than or equal to 2.
d) Calculate the value of the expression $$
f(2016,1)+f(2016,4)+f(2016,7)+\ldots+f(2016,4027)+f(2016,4030)
$$
a) Display all 6 ways to distribute 4 chocolates to 3 children with each one earning a maximum of two chocolates.
b) Considering 2015 children, check that $f(2015, k)=0$ for all $k$ greater than or equal to an appropriate value.
c) Shows that the equation $$
f(2016, k)=f(2015, k)+f(2015, k-1)+f(2015, k-2)
$$ is true for all $k$ positive integer greater than or equal to 2.
d) Calculate the value of the expression $$
f(2016,1)+f(2016,4)+f(2016,7)+\ldots+f(2016,4027)+f(2016,4030)
$$