Download Formulate a combinatorial problem that leads to the following

Question: Formulate a combinatorial problem that leads to the following generating function:
(1  x  x 2 )(1  x 2  x 4  x 6 )(1  x 2  x 4  )( x  x 2  x 3  )
My solution:
Find the number of n-combinations of apples, bananas, oranges, and pears. In each combination,
the number of apples is no more than 2, the number of bananas is no more than 6 and is even, the
number of oranges is even, and there is at least one pear. We note that the problem is equivalent to
finding the number hn of nonnegative integral solutions of the equation e1  e2  e3  e4  n ,
where 0  e1  2, e2 {0,2,4,6}, e3 is even, and e4  1. We create one factor for each type of fruit,
where the exponents are the allowable numbers in the n-combinations for that type of fruit, then
we have g ( x)  (1  x  x )(1  x  x  x )(1  x  x  )( x  x  x  ) .
2
2
4
6
2
4
2
3
1 x3
1  x8
1
, 1 x2  x4  x6 
, 1 x2  x4  
, and
2
1 x
1 x
1 x2
x
x  x2  x3   
1 x
1 x  x2 
So, g ( x) 
1  x3 1  x8 1
x
x(1  x 3 )(1  x 8 )
, and the coefficients in the Taylor

1  x 1  x 2 1  x 2 1  x (1  x) 2 (1  x 2 ) 2
series for this rational function count the number hn of combinations of the type considered.