The problem is more easily read and less confusing if you will use
number words for the numerals and numerals for the number word. That is,
rewrite the problem with
the numeral "7" instead of the word "seven",
the numeral "9" instead of the word "nine",
the word "five" instead of the numeral "5",
the word "four" instead of the numeral "4",
like this:
I have 5 seven-peso stamps and 4 nine-peso stamps. Using one or more of these
stamps, find as many different amounts of postage that you can mail with them.
There are 5 seven-peso stamps so there are 6 ways to choose the number of
seven-peso stamps. The 6 ways are 0, 1, 2, 3, 4, or 5 seven-peso stamps because
we must include the use of NONE of them.
There are 4 nine-peso stamps so there are 5 ways to choose the number of
nine-peso stamps. The 5 ways are 0, 1, 2, 3, or 4 nine-peso stamps because as
above we must include the use of NONE of them.
That would be a total of 6×5 or 30. And there cannot be two ways to make any
sum because seven and nine are relatively prime and there aren't enough of
either stamp to make their least common multiple.
However from the 30 we must subtract the case where we use 0 seven-peso stamps
and also 0 nine-peso stamps. 0 is not an amount of postage.
Answer: 29 amounts of postage.
Edwin
