Fruit basket
Task number: 3866
In how many ways can a basket be filled with \(n\) fruits under the following (somewhat unusual) conditions?
- The number of apples must be even,
- the number of bananas is divisible by five,
- there are at most four oranges in the basket
- and one or no pear.
Solution
Generating functions for the numbers of individual types of fruits (a,b,o,p) are
\(g_a(x)=x^0+x^2+x^4+… =\frac1{1-x^2}\),
\(g_b(x)=x^0+x^5+x^{10}+… =\frac1{1-x^5}\),
\(g_o(x)=x^0+x^1+x^2+x^3+x^4=\frac{1-x^5}{1-x}\) a
\(g_p(x)=x^0+x^1=1+x\).
The product of these functions gives the resulting generating function
\(g(x)=g_a(x)g_b(x)g_o(x)g_p(x)= \frac1{1-x^2}\cdot\frac1{1-x^5}\cdot\frac{1-x^5}{1-x}(1+x)= \frac1{(1-x)^2}=1x^0+2x^1+3x^2+4x^3+5x^4+…\).
Answer
A basket with \( n \) fruits can be assembled in \( n+1 \) ways.
