With the requirement of 1 40˘ stamp, that reduces the problem to
"How many different ways can she make 23˘ postage using a combination
of 5˘, 3˘, and 2˘ stamps?"
For this problem, we need to know the rules of operations with even
and odd numbers, sometimes known as the "parity rules":
EVEN+EVEN=EVEN EVEN×EVEN=EVEN
EVEN+ODD=ODD EVEN×ODD=EVEN
ODD+ODD=EVEN ODD×ODD=ODD
I. If she doesn't use any 5 cent stamps, then:
To make 23˘ with 3˘ and 2˘ stamps only, since 23 is odd, it can
only be written as an even number plus an odd number.
Any whole number of 2˘ stamps will produce an even number of
cents. Therefore to get an odd number to add to that even number to
make 23, she must use an odd number of 3˘ stamps. (That's because
an even number of 3˘ stamps would be an even number of cents.)
There are four odd numbers of 3˘ stamps possible (1,3,5, and 7).
That gives 4 ways since she can always make up the remaining even
part of 23˘ with 2˘ stamps. So that's 4 ways.
2. If she uses exactly 1 5˘ stamp, then the problem reduces to
"How many different ways can she make 18˘ postage using a combination
of 3˘ and 2˘ stamps?"
18 is even. Any number of 2˘ stamps is even, so the 3˘ stamps
must also contribute an even number of cents, so she can only
use an even number of 3˘ stamps. There are four even numbers
of 3˘ stamps possible (0,2,4, and 6). That gives 4 ways since she can
always make up the remaining even part of 23˘ with 2˘ stamps. So
that's 4 more ways.
3. If she uses exactly 2 5˘ stamp, then the problem reduces to
"How many different ways can she make 13˘ postage using a combination
of 3˘ and 2˘ stamps?"
13 is odd. Any number of 2˘ stamps is even, so the 3˘ stamps
must contribute an odd number of cents, so she can only
use an odd number of 3˘ stamps. There are two odd numbers
of 3˘ stamps possible (1 and 3). That gives 2 ways since she can
always make up the remaining even part of 23˘ with 2˘ stamps. So
that's 2 more ways.
4. If she uses exactly 3 5˘ stamp, then the problem reduces to
"How many different ways can she make 8˘ postage using a combination
of 3˘ and 2˘ stamps<"
8 is even. Any number of 2˘ stamps is even, so the 3˘ stamps
must contribute an even number of cents, so she can only
use an even number of 3˘ stamps. There are two even numbers
of 3˘ stamps possible (0 and 2). That gives us 2 ways since she can
always make up the remaining even part of 23˘ with 2˘ stamps. So
that's 2 more ways.
5. If she uses exactly 3 5˘ stamp, then the problem reduces to
"How many different ways can she make 3˘ postage using a combination
of 3˘ and 2˘ stamps?"
Obviously there is only 1 way -- 1 3˘ stamp and no 2˘ stamps.
That's 1 more way.
So the answer, adding the numbers of cases from above, is
4+4+2+2+1 = 13 ways.
Edwin