THE FOUR POSTAGE STAMPS.
(
Combination and Group Problems)
+---+----+----+----+
| 1 | 2 | 3 | 4 |
+---+----+----+----+
| 5 | 6 | 7 | 8 |
+---+----+----+----+
| 9 | 10 | 11 | 12 |
+---+----+----+----+
"It is as easy as counting," is an expression one sometimes hears. But
mere counting may be puzzling at times. Take the following simple
example. Suppose you have just bought twelve postage stamps, in this
form--three by four--and a friend asks you to oblige him with four
stamps, all joined together--no stamp hanging on by a mere corner. In
how many different ways is it possible for you to tear off those four
stamps? You see, you can give him 1, 2, 3, 4, or 2, 3, 6, 7, or 1, 2, 3,
6, or 1, 2, 3, 7, or 2, 3, 4, 8, and so on. Can you count the number of
different ways in which those four stamps might be delivered? There are
not many more than fifty ways, so it is not a big count. Can you get the
exact number?
Answer:
Referring to the original diagram, the four stamps may be given in the
shape 1, 2, 3, 4, in three ways; in the shape 1, 2, 5, 6, in six ways;
in the shape 1, 2, 3, 5, or 1, 2, 3, 7, or 1, 5, 6, 7, or 3, 5, 6, 7, in
twenty-eight ways; in shape 1, 2, 3, 6, or 2, 5, 6, 7, in fourteen ways;
in shape 1, 2, 6, 7, or 2, 3, 5, 6, or 1, 5, 6, 10, or 2, 5, 6, 9, in
fourteen ways. Thus there are sixty-five ways in all.
