Question

A fruit shop has \(4\) different types of bananas. The number of ways in which \(12\) bananas can be bought with at least one banana from each type, is .

Hint:

For distributing \(n\) identical objects among \(r\) groups with at least one in each group, use \(\binom{n-1}{r-1}\).

Answer 1

Correct Answer: 165

Step 1: Turn it into an equation.
Let $x_1,x_2,x_3,x_4$ be the counts of each banana type. We need $x_1+x_2+x_3+x_4=12$.

Step 2: Use the 'at least one' rule.
Each $x_i\ge 1$. Set $y_i=x_i-1$, so each $y_i\ge 0$.

Step 3: New equation.
\[ y_1+y_2+y_3+y_4=8 \]

Step 4: Count with stars and bars.
The number of non negative solutions is $\binom{8+3}{3}=\binom{11}{3}$.

Step 5: Compute.
\[ \binom{11}{3}=\frac{11\cdot 10\cdot 9}{6}=165 \]
\[ \boxed{165} \]