Question

There are four different types of bananas. In how many ways can 12 children select bananas so that at least one child selects different types of bananas?

Hint:

For “at least one” type problems, it is often easier to count the total possibilities and subtract the unwanted cases using the complement principle.

Answer 1

Each child can choose any one of the 4 different types of bananas.
Step 1: Total number of ways without restriction.
Since each of the 12 children has 4 choices independently,
\[ \text{Total ways} = 4^{12} \] Step 2: Count the number of ways where all children choose the same type.
If all children choose the same type of banana,
There are only 4 possible choices (all choose type 1, or type 2, or type 3, or type 4).
So, number of such ways = 4.
Step 3: Use complementary counting.
We want at least one child to choose a different type,
So subtract the cases where all choose the same type.
\[ \text{Required ways} = 4^{12} - 4 \] Step 4: Final expression.
\[ \boxed{4^{12} - 4} \]