Question

12. How many different combinations of pennies, nickels, dimes, quarters, and half dollars can a piggy bank contain if it has 20 coins in it?

Short answer

Thus there are $10,626$ways a piggy bank can contain pennies, nickels, dimes, quarters and half dollars if it contains twenty coins.

Step-by-step solution

Step 1: Definitions

Definition of Permutation (Order is important)

No repetition allowed:$P(n,r)=\frac{n!}{(n-r)!}$

Repetition allowed:$n^T$

Definition of combination (order is important)

No repetition allowed:$C(n,r)=\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{r!(n-r)!}$

Repetition allowed:$C(n+r-1,r)=\left(\begin{array}{c}n+r-1\\ r\end{array}\right)=\frac{(n+r-1)!}{r!(n-1)!}$

with$n!=n(n-1)\cdot .....\cdot 2\cdot 1$

Solutions

The order of the elements does not matter (as a different order result in the same number of pennies), thus we need to use the definition of combination.

We are interested in selecting r = 20 elements from a set with n = 5 elements.(pennies, nickels, dimes, quarters and half dollars)

Repetition of elements is allowed

$\begin{array}{r}C(n+r-1,r)=C(5+20-1,20)\\ C(24,20)=\frac{24!}{20!(24-20)!}=\frac{24!}{20!4!}=10,626\end{array}$

Thus there are ways a piggy bank can contain pennies, nickels, dimes, quarters and half dollars if it contains twenty coins.