## Values From Three Coins

How many possible values can there be for three coins selected from among pennies, nickels, dimes, and quarters.
Source: NCTM Mathematics Teacher, August 2006

Solution
a) All three coins are the same
$1\:1\:1$
$5\:5\:5$
$10\:10\:10$
$25\:25\:25$
Number of ways = $4$
b) Two of the coins are the same
$1\:1\:-$
$-\:1\:1$
$1\:-\:1$
$\cdots$
Number of ways = $3\times 4=12$
c) All three coins are different
Number of ways = $\dbinom {4}{3}=4$
Total number of ways = $4+12+4=20$
If we check the values of these $20$ ways, the values are all different. There are $20$ possible values when three coins are selected from among pennies, nickels, dimes, and quarters.

Answer: $20$